A bridge modal identification method based on frequency domain analysis

Abstract

The application discloses a bridge modal identification method based on frequency domain analysis, comprising the following steps: collecting line-of-sight distance change data of a plurality of monitoring points on a target bridge, and performing mean removal processing on the distance change data to obtain preprocessed monitoring data; calculating cross-correlation functions between the monitoring points by using fast Fourier transform (FFT), constructing an autocorrelation function matrix, and performing Fourier transform to obtain a power spectral density matrix; performing singular value decomposition on the power spectral density matrix at each frequency point to obtain a singular value curve, performing peak value detection based on the maximum singular value curve, and identifying a peak frequency point of a candidate mode; taking a mode shape at the peak frequency point as a reference mode shape, extracting a continuous frequency interval with a correlation higher than a preset threshold as a modal region; and estimating inherent frequencies, mode shapes and damping ratios of the target bridge structure based on the singular value decomposition result and the modal region. The application realizes high-precision and automatic bridge modal identification.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring technology, and in particular to a method for bridge modal identification based on frequency domain analysis. Background Technology

[0002] As a key component of transportation infrastructure, the structural safety of bridges is directly related to the safety of people's lives and property. With the continuous increase in traffic volume and the lengthening of bridge service life, the importance of structural health monitoring is becoming increasingly prominent. Traditional bridge inspection methods mainly rely on manual inspections and periodic checks, which have drawbacks such as long inspection cycles, high costs, and strong subjectivity, making it difficult to achieve real-time and continuous assessment of the bridge structural condition.

[0003] In recent years, structural health monitoring technology has been continuously developing, among which modal identification, as a core method for analyzing structural dynamic characteristics, has received widespread attention. Existing bridge modal identification technologies mainly include the following categories: (1) Time-domain methods: such as random subspace identification method. This method directly uses time history response data for parameter identification, but the computational complexity is high, it is sensitive to noise, and the identification results are not stable.

[0004] (2) Frequency domain methods: such as peak picking method. This method identifies modal frequencies by analyzing the peak values ​​of the power spectral density curve, but the frequency resolution is limited and it is difficult to effectively identify dense modes.

[0005] (3) Time-frequency analysis methods: such as wavelet transform. This method can provide both time-domain and frequency-domain information at the same time, but the parameter selection is difficult and the computation is large, making it unsuitable for long-term online monitoring.

[0006] The aforementioned existing technologies generally suffer from the following problems in application: (1) A large number of sensors need to be deployed, resulting in high system costs and complex installation and maintenance; (2) It has strict requirements for monitoring equipment and is difficult to apply to new non-contact monitoring methods such as radar; (3) The accuracy of modal parameter identification is insufficient, especially the identification error of damping ratio is relatively large; (4) The data processing process is complex and relies on manual intervention, making it difficult to achieve automated online monitoring.

[0007] On the other hand, MIMO radar, as a novel non-contact monitoring device, has advantages such as high precision, simultaneous monitoring of multiple measurement points, and convenient deployment, and is gradually being introduced into the field of bridge structural health monitoring. However, MIMO radar can only monitor line-of-sight deformation, that is, displacement changes in the direction of the line connecting the monitoring point and the radar, and cannot directly obtain the multi-directional vibration response of the structure, which limits the application of traditional modal identification methods.

[0008] Therefore, there is an urgent need for a bridge modal identification method that can adapt to the characteristics of MIMO radar monitoring, and has high precision, high efficiency and automated processing capabilities, in order to solve the problems existing in the current technology. Summary of the Invention

[0009] This invention proposes a bridge modal identification method based on frequency domain analysis, which solves the problems existing in the prior art and is particularly suitable for modal parameter identification of MIMO radar monitoring data.

[0010] To achieve the above objectives, the present invention provides a bridge modal identification method based on frequency domain analysis, comprising: Data on the change in line-of-sight distance at several monitoring points on the target bridge are collected, and the distance change data is processed to remove the mean, resulting in preprocessed monitoring data. Based on the preprocessed monitoring data, the cross-correlation function between each monitoring point is calculated using Fast Fourier Transform (FFT), an autocorrelation function matrix is ​​constructed, and the autocorrelation function matrix is ​​subjected to Fourier transform to obtain the power spectral density matrix. Singular value decomposition is performed on the power spectral density matrix to obtain singular value curves. Peak detection is then performed based on the maximum singular value curves to identify the peak frequency points of candidate modes. Using the mode shape at the peak frequency point as the reference mode shape, the correlation between the peak frequency point and the mode shape of the neighboring frequency point is calculated using the Modal Confidence Criterion (MAC), and the continuous frequency interval with a correlation higher than a preset threshold is extracted as the modal region. Based on the singular value decomposition results and the modal region, the natural frequencies, mode shapes, and damping ratios of the target bridge structure are estimated.

[0011] Preferably, constructing the autocorrelation function matrix includes: The preprocessed monitoring data is zero-filled and subjected to FFT transformation to obtain the frequency domain signal; Calculate the cross-power spectrum between the frequency domain signals of any two monitoring points; Perform an inverse FFT transform on the cross-power spectrum to obtain the corresponding cross-correlation function; A three-dimensional autocorrelation function matrix is ​​constructed based on the cross-correlation function among all monitoring points.

[0012] Preferably, the power spectral density matrix is ​​obtained by performing a Fourier transform on the autocorrelation function matrix, specifically as follows: ; In the formula, Here is the power spectral density matrix, ω=2π f Angular frequency, f For frequency, The autocorrelation function matrix, Due to time lag, The sampling time interval is FFT, which stands for Fast Fourier Transform operator.

[0013] Preferably, the power spectral density matrix G(ω) is a Hermitian matrix, satisfying G( ω )=G H ( ω ), H Indicates conjugate transpose; The diagonal elements of the Hermitian matrix represent the self-power spectral density, while the off-diagonal elements represent the cross-power spectral density.

[0014] Preferably, the power spectral density matrix at each frequency point is decomposed into singular values ​​as follows: ; In the formula, For the first k Frequency points The power spectral density matrix at that location; For the first k angular frequency at each frequency point; V( is a left singular vector matrix;) ωk ) is a right singular vector matrix; H Indicates conjugate transpose; Nf This represents the number of frequency points.

[0015] Preferably, peak detection is performed based on the maximum singularity curve, including: Calculate the mean and standard deviation of the maximum singularity curve, and set a dynamic threshold; Identify peak points on the maximum singularity curve that exceed the dynamic threshold; Adjacent peak points that are spaced less than a preset interval on the frequency axis are merged.

[0016] Preferably, extracting continuous frequency intervals with correlations higher than a preset threshold as modal regions includes: The vibration mode at the peak frequency point is set as follows: Within a preset bandwidth range, calculate the MAC value of the mode shape and the reference mode shape corresponding to each frequency point; The continuous frequency range where the MAC value is greater than the preset threshold is defined as the modal region of the corresponding order; Before extracting the next modal region, remove frequency points that are already occupied by the current mode.

[0017] Preferably, the natural frequency of the target bridge structure is directly determined by the peak frequency, specifically: ; In the formula, For the natural frequency, This is the peak frequency.

[0018] Preferably, the mode shape of the target bridge structure is directly given by the first-order singular vector at the peak frequency point, specifically: ; In the formula, It is the mode shape; The power spectral density matrix G( fp At peak frequency fp After performing singular value decomposition at point , the left singular vector matrix The first column.

[0019] Preferably, the identification of the damping ratio of the target bridge structure includes: Using the modal region as the bandwidth and the mode shape of the corresponding order mode as the weight, the single-degree-of-freedom SDOF power spectrum is extracted; Perform an inverse Fourier transform on the SDOF power spectrum to obtain the single-degree-of-freedom autocorrelation function of the corresponding mode. Based on the exponential decay characteristics of the single-degree-of-freedom autocorrelation function, the damping ratio of the corresponding mode can be identified by logarithmic fitting or the half-power bandwidth method.

[0020] Compared with the prior art, the present invention has the following advantages and technical effects: (1) Applicable to radar monitoring: Specifically designed for the characteristics of MIMO radar that can only monitor line-of-sight deformation, making full use of radar monitoring data.

[0021] (2) High-precision identification: Through SVD decomposition and MAC criterion, the modal parameters are accurately identified, avoiding the aliasing of dense modes.

[0022] (3) Automated processing: The entire process can be executed automatically without human intervention, and is suitable for long-term online monitoring.

[0023] (4) High computational efficiency: FFT is used to accelerate the calculation of correlation functions, and eigenvalue decomposition replaces SVD, which significantly improves computational efficiency.

[0024] (5) Accurate damping ratio identification: The SDOF power spectrum method is adopted to improve the identification accuracy of damping ratio.

[0025] (6) Non-contact monitoring: Based on radar monitoring, non-contact bridge health monitoring is realized, which is convenient to deploy and has low cost. Attached Figure Description

[0026] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a time history curve of the distance change of each monitoring point in an embodiment of the present invention; Figure 2 This is a graph showing the autocorrelation function of each monitoring point in an embodiment of the present invention; Figure 3 This is a comparison chart of the power spectral density at various monitoring points in an embodiment of the present invention; Figure 4 This is a schematic diagram of the singular value decomposition results in an embodiment of the present invention; Figure 5 This is a schematic diagram of the outlier peak detection results in an embodiment of the present invention; Figure 6 This is a flowchart of the EFDD algorithm according to an embodiment of the present invention; Figure 7 This is a scatter plot of the zero-crossing points of the first-order mode in an embodiment of the present invention. Figure 8 This is a scatter plot of the zero-crossing points of the second-order modes in an embodiment of the present invention. Figure 9 This is a schematic diagram of the first-order modal autocorrelation function (inverse Fourier transform) according to an embodiment of the present invention; Figure 10 This is a schematic diagram of the second-order modal autocorrelation function (inverse Fourier transform) according to an embodiment of the present invention. Detailed Implementation

[0027] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0028] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0029] This embodiment proposes a bridge modal identification method based on frequency domain analysis, including: Data on the change in line-of-sight distance at several monitoring points on the target bridge are collected, and the distance change data is processed to remove the mean, resulting in preprocessed monitoring data. Based on the preprocessed monitoring data, the cross-correlation function between each monitoring point is calculated using Fast Fourier Transform (FFT), an autocorrelation function matrix is ​​constructed, and the autocorrelation function matrix is ​​subjected to Fourier transform to obtain the power spectral density matrix. Singular value decomposition is performed on the power spectral density matrix at each frequency point to obtain singular value curves. Peak detection is then performed based on the maximum singular value curve to identify the peak frequency points of candidate modes. Using the mode shape at the peak frequency point as the reference mode shape, the correlation between the peak frequency point and the mode shape of the neighboring frequency point is calculated using the Modal Confidence Criterion (MAC), and the continuous frequency interval with a correlation higher than a preset threshold is extracted as the modal region. Based on the singular value decomposition results and the modal region, the natural frequencies, mode shapes, and damping ratios of the target bridge structure are estimated.

[0030] Furthermore, line-of-sight distance variation data from several monitoring points on the target bridge are collected, and the distance variation data is processed to remove the mean, resulting in preprocessed monitoring data, including: Distance variation data of bridge monitoring points are collected using MIMO radar. Assume the radar monitors M monitoring points, and the sampling frequency is... The collection time is T Then the distance change data matrix is ​​obtained. : ; In the formula, The number of sampling points. Indicates the first i Each measuring point is at The change in distance over time.

[0031] Perform mean-removal processing on the data: ; In the formula, For the first i Each measuring point is at tj The data after removing the mean at each time step. For the first i The mean of the distance change data of each measuring point.

[0032] Further, the autocorrelation function matrix is ​​constructed, including: The preprocessed monitoring data is zero-filled and subjected to FFT transformation to obtain the frequency domain signal; Calculate the cross-power spectrum between the frequency domain signals of any two monitoring points; Perform an inverse FFT transform on the cross-power spectrum to obtain the corresponding cross-correlation function; A three-dimensional autocorrelation function matrix is ​​constructed based on the cross-correlation function among all monitoring points.

[0033] Specifically, the autocorrelation function matrix is ​​calculated using the FFT method. For the mean-removed data, the cross-correlation function is calculated: ; In the formula, For the first i The and the first j The cross-correlation function between measurement points For the firsti Each measuring point is at tk The mean-removed data at time point, For the first j Each measuring point is at tk + τ The mean-removed data at each time point; tk For the first k Each sampling time; τ This is due to a time lag.

[0034] To improve computational efficiency, the FFT method is used to calculate the cross-correlation function: 1. Perform zero-filling and FFT transformation on the signal at each measurement point: ; In the formula, For the first i Frequency domain signal at each measurement point The time-domain signal after zero padding. The length of the FFT is greater than [value missing]. The smallest power of 2, This represents the maximum lag time.

[0035] 2. Calculate the cross-power spectrum: ; In the formula, For the first i The and the first j Cross-power spectrum between measurement points For the first j Frequency domain signal at each measurement point The complex conjugate, For the first i Frequency domain signal at each measurement point.

[0036] 3. Obtain the cross-correlation function through inverse FFT: ; In the formula, For the first i The and the first j Cross-correlation function between measurement points; Construct the autocorrelation function matrix: ; In the formula, The autocorrelation function matrix, M This represents the number of measurement points.

[0037] Further, a Fourier transform is performed on the autocorrelation function matrix to obtain the power spectral density matrix, specifically: Performing an FFT on the autocorrelation function matrix yields the power spectral density matrix: ; In the formula, Here is the power spectral density matrix. The autocorrelation function matrix, The sampling time interval, ω is the angular frequency.

[0038] For a Hermitian matrix G(ω), the following holds: ; Among them, diagonal elements The power spectral density is the off-diagonal element. This represents the cross-power spectral density.

[0039] Furthermore, the power spectral density matrix at each frequency point is decomposed into singular values ​​as follows: ; In the formula, For the first k Frequency points The power spectral density matrix at that location; For the first k angular frequency at each frequency point; V( is a left singular vector matrix;) ωk ) is a right singular vector matrix; H Indicates conjugate transpose; Nf This represents the number of frequency points.

[0040] Specifically, due to It is a Hermitian positive semi-definite matrix, and eigenvalue decomposition can be used instead of SVD, which is more computationally efficient. ; Among them, the singular value is the square root of the eigenvalue. , These are the eigenvalues.

[0041] Furthermore, peak detection is performed based on the maximum singularity curve, including: Calculate the mean and standard deviation of the maximum singularity curve, and set a dynamic threshold; Identify peak points on the maximum singularity curve that exceed the dynamic threshold; Adjacent peak points that are spaced less than a preset interval on the frequency axis are merged.

[0042] Specifically, peak detection based on the maximum singularity curve includes: Calculate threshold ; In the formula, and These are the mean and standard deviation, respectively. Maximum singular value The mean, Maximum singular value The standard deviation.

[0043] Extract peak points that exceed the threshold: ; In the formula, Peak frequency, For the first k One frequency point; For the first k The maximum singular value at each frequency point; Threshold is a dynamic threshold. This is the set of peak frequencies.

[0044] Adjacent peak values ​​are merged (frequency points with an interval of less than 5 are considered as the same peak value).

[0045] Furthermore, the frequency bandwidth of each mode is determined using the Modal Confidence Criterion (MAC): At peak frequency mode shape at For reference mode shape; Within the preset bandwidth, calculate the MAC value of the mode shape and the reference mode shape for each frequency point: ; In the formula, For frequency f The mode shape at that point, Peak frequency point fp Reference mode shape at that location, H Indicates conjugate transpose; |·| Represents the modulus of a complex number; This represents the modal confidence criterion value.

[0046] Extract the continuous frequency region with MAC > 0.9 as the modal region; Remove already occupied frequency points to avoid modal aliasing.

[0047] Furthermore, based on the singular value decomposition results and the determined modal regions, the modal parameters of each order are estimated: (1) Identification of inherent frequency: The natural frequency is given directly from the peak frequency: ; In the formula, This is the natural frequency.

[0048] (2) Mode identification: The mode shape is given by singular value decomposition.

[0049] At peak frequency At that point, the power spectral density matrix The first singular vector is the mode shape of that order: ; In the formula, For vibration modes, The power spectral density matrix G( fp At peak frequency fp After performing singular value decomposition at point , the left singular vector matrix The first column.

[0050] Furthermore, the identification of the damping ratio of the target bridge structure includes: Using the modal region as the bandwidth and the mode shape of the corresponding order mode as the weight, the single-degree-of-freedom SDOF power spectrum is extracted; Perform an inverse Fourier transform on the SDOF power spectrum to obtain the single-degree-of-freedom autocorrelation function of the corresponding mode. Based on the exponential decay characteristics of the single-degree-of-freedom autocorrelation function, the damping ratio of the corresponding mode can be identified by logarithmic fitting or the half-power bandwidth method.

[0051] Specifically, based on the EFDD method, the damping ratio is identified through the single-degree-of-freedom (SDOF) power spectrum: Step a: Extract the SDOF power spectrum: Within a defined modal region, with mode shapes As weights, extract the SDOF power spectrum: ; in, The modal region boundary is determined by the MAC criterion. The left boundary frequency, The right boundary frequency, This is a single-degree-of-freedom power spectrum; It is a mode shape; The power spectral density matrix; H This indicates the conjugate transpose.

[0052] Step b: Calculate the autocorrelation function: Performing an inverse FFT on the SDOF power spectrum yields the autocorrelation function of this mode: ; In the formula, It is a single-degree-of-freedom autocorrelation function; This represents the power spectrum of a single degree of freedom; IFFT stands for Inverse Fast Fourier Transform operator. This is due to a time lag.

[0053] Step c: Damping ratio estimation: The autocorrelation function exhibits exponential decay: ; In the formula,A Amplitude; ξ The damping ratio; It is the damped natural circular frequency; φ The phase angle; e is the base of the natural logarithm; τ This is due to a time lag.

[0054] The damping ratio can be identified by logarithmic fitting of the autocorrelation function envelope or by the half-power bandwidth method. ; In the formula, c The exponential decay coefficient is obtained from the fitting.

[0055] To verify the effectiveness of the bridge modal identification method based on frequency domain analysis proposed in this embodiment, simulation experiments were conducted.

[0056] The simulation settings are as follows: Measurement point configuration: 8 measurement points (N_283, N_303, N_323, N_343, N_366, N_386, N_743, N_763); Sampling frequency: 50Hz (sampling time interval ∆t=0.02s); Data length: determined based on the actual data; Structural model: Simulates the first few modes of the bridge; Noise level: Add an appropriate level of Gaussian white noise to simulate the actual measurement environment.

[0057] The simulation steps include: Step 1, Raw Data: Figure 1 The time history curves showing the distance changes at eight measuring points are displayed. From Figure 1 As can be seen, the vibration responses at each measuring point have a good correlation, reflecting the overall vibration characteristics of the bridge.

[0058] Step 2: Calculation of the autocorrelation function matrix: The autocorrelation function matrix was calculated using the FFT method, with the maximum lag set to 3000 points (corresponding to 60 seconds). Figure 2 The autocorrelation function curves for each measuring point are displayed.

[0059] Step 3: Calculation of power spectral density matrix: The power spectral density matrix is ​​obtained by performing an FFT transformation on the autocorrelation function matrix.

[0060] The frequency resolution is: ; The frequency range is 0 to 25 Hz (Nyquist frequency). Figure 3The power spectral density curves for each measurement point are shown. Figure 3 Multiple peaks can be clearly observed, corresponding to different modes of the structure.

[0061] Step 4, Singular Value Decomposition: Eigenvalue decomposition is performed on the power spectral density matrix at each frequency point to obtain singular value curves. Figure 4 This shows how the first two singular values ​​change with frequency.

[0062] Step 5, Peak Detection: Peak detection is performed based on the maximum singularity curve.

[0063] Set the threshold to the mean plus twice the standard deviation: ; Multiple significant peaks were detected, corresponding to different modes of the structure. The peak detection results are as follows: Figure 5 As shown.

[0064] Step 6: Modal region extraction: Using each detected peak frequency as a reference, the modal region is extracted using the MAC criterion. The MAC threshold is set to 0.9, and the bandwidth is ±5Hz. By calculating the MAC value of each frequency point relative to the reference mode shape, continuous frequency regions with MAC > 0.9 are extracted.

[0065] Step 7, Modal parameter identification: Inherent frequency identification: The inherent frequency is given directly from the peak frequency. Table 1 provides a complete comparison between the actual frequency and the identified frequency.

[0066] Table 1 As can be seen from Table 1, the error between the identified frequency and the actual frequency is within 0.01Hz, and the relative error is less than 0.1%, indicating that the method in this embodiment has extremely high frequency identification accuracy.

[0067] Through the above simulation verification, the bridge modal identification method based on frequency domain analysis proposed in this embodiment exhibits the following advantages: 1. High frequency identification accuracy: The inherent frequency identification error is less than 0.1%, meeting the requirements of high-precision monitoring.

[0068] 2. Accurate mode shape identification: MAC value is greater than 0.95, and the mode shape matches the theoretical value well.

[0069] 3. Reliable damping ratio identification: The damping ratio identification error is less than 2%, which meets the requirements of engineering applications.

[0070] 4. Excellent mode separation: It effectively separates dense modes using the MAC criterion, avoiding mode aliasing.

[0071] 5. Strong noise resistance: It can still accurately identify modal parameters under appropriate noise levels.

[0072] In summary, the method of this embodiment can effectively identify the modal parameters of bridges, and is particularly suitable for processing MIMO radar monitoring data. It has the advantages of high precision, high efficiency and high degree of automation.

[0073] Figure 6 It demonstrates the complete process of the EFDD algorithm, from data acquisition to modal parameter identification. Figure 7 A scatter plot of the zero-crossing points of the first-order mode is shown to verify the continuity of the mode shape. Figure 8 The zero-crossing point scatter plot of the second-order mode is shown. Figure 9 The first-order modal autocorrelation function obtained by inverse Fourier transform is shown and used for damping ratio identification. Figure 10 The autocorrelation function of the second-order mode is shown.

[0074] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A bridge modal identification method based on frequency domain analysis, characterized in that, include: Data on the change in line-of-sight distance at several monitoring points on the target bridge are collected, and the distance change data is processed to remove the mean, resulting in preprocessed monitoring data. Based on the preprocessed monitoring data, the cross-correlation function between each monitoring point is calculated using Fast Fourier Transform (FFT), an autocorrelation function matrix is ​​constructed, and the autocorrelation function matrix is ​​subjected to Fourier transform to obtain the power spectral density matrix. Singular value decomposition is performed on the power spectral density matrix to obtain singular value curves. Peak detection is then performed based on the maximum singular value curves to identify the peak frequency points of candidate modes. Using the mode shape at the peak frequency point as the reference mode shape, the correlation between the peak frequency point and the mode shape of the neighboring frequency point is calculated using the Modal Confidence Criterion (MAC), and the continuous frequency interval with a correlation higher than a preset threshold is extracted as the modal region. Based on the singular value decomposition results and the modal region, the natural frequencies, mode shapes, and damping ratios of the target bridge structure are estimated.

2. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, Constructing the autocorrelation function matrix includes: The preprocessed monitoring data is zero-filled and subjected to FFT transformation to obtain the frequency domain signal; Calculate the cross-power spectrum between the frequency domain signals of any two monitoring points; Perform an inverse FFT transform on the cross-power spectrum to obtain the corresponding cross-correlation function; A three-dimensional autocorrelation function matrix is ​​constructed based on the cross-correlation function among all monitoring points.

3. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, Performing a Fourier transform on the autocorrelation function matrix yields the power spectral density matrix, specifically: ； In the formula, Here is the power spectral density matrix, ω=2π f Angular frequency, f For frequency, The autocorrelation function matrix, Due to time lag, The sampling time interval is FFT, which stands for Fast Fourier Transform operator.

4. The bridge modal identification method based on frequency domain analysis according to claim 3, characterized in that, The power spectral density matrix G(ω) is a Hermitian matrix, satisfying G( ω )=G H ( ω ), H Indicates conjugate transpose; The diagonal elements of the Hermitian matrix represent the self-power spectral density, while the off-diagonal elements represent the cross-power spectral density.

5. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, The power spectral density matrix at each frequency point is decomposed into singular values ​​as follows: ； In the formula, For the first k Frequency points The power spectral density matrix at that location; For the first k angular frequency at each frequency point; V( is a left singular vector matrix;) ωk ) is a right singular vector matrix; H Indicates conjugate transpose; Nf This represents the number of frequency points.

6. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, Peak detection based on the maximum singularity curve includes: Calculate the mean and standard deviation of the maximum singularity curve, and set a dynamic threshold; Identify peak points on the maximum singularity curve that exceed the dynamic threshold; Adjacent peak points that are spaced less than a preset interval on the frequency axis are merged.

7. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, Extract continuous frequency intervals with correlations higher than a preset threshold as modal regions, including: The vibration mode at the peak frequency point is set as follows: Within a preset bandwidth range, calculate the MAC value of the mode shape and the reference mode shape corresponding to each frequency point; The continuous frequency range where the MAC value is greater than the preset threshold is defined as the modal region of the corresponding order; Before extracting the next modal region, remove frequency points that are already occupied by the current mode.

8. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, The natural frequency of the target bridge structure is directly determined by its peak frequency, specifically: ； In the formula, For the natural frequency, This is the peak frequency.

9. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, The mode shape of the target bridge structure is directly given by the first-order singular vector at the peak frequency point, specifically: ； In the formula, It is the mode shape; The power spectral density matrix G( fp At peak frequency fp After performing singular value decomposition at point , the left singular vector matrix The first column.

10. The bridge modal identification method based on frequency domain analysis according to claim 1, characterized in that, The identification of the damping ratio of the target bridge structure includes: Using the modal region as the bandwidth and the mode shape of the corresponding order mode as the weight, the single-degree-of-freedom SDOF power spectrum is extracted; Perform an inverse Fourier transform on the SDOF power spectrum to obtain the single-degree-of-freedom autocorrelation function of the corresponding mode. Based on the exponential decay characteristics of the single-degree-of-freedom autocorrelation function, the damping ratio of the corresponding mode can be identified by logarithmic fitting or the half-power bandwidth method.

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