Automatic extraction method for fundamental frequency of steel-structure high-speed railway bridge under operation condition

Abstract

The invention provides a method for automatically extracting the fundamental frequency of a steel-structure high-speed railway bridge under an operation condition, which comprises the steps of sensor layout and response signal acquisition, working condition identification and quasi-dead load data fragment segmentation, multi-measuring-point coherent weighted stacking, rough frequency locking, fundamental frequency fine identification, statistical fusion output and the like. The manual excitation test is carried out without interrupting the transportation of the train, the extracted data completely depends on the data of the operating train to realize automatic monitoring, and the applicability is wide. In the signal extraction process, through the coherent weighted stacking technology, the signal extraction problem that the steel structure bridge is small in vibration and large in field noise is effectively solved, and the anti-interference capability is high. In combination with quasi-dead load fragment screening and high-resolution spectrum estimation, strong excitation energy of live loads is utilized, frequency errors caused by non-stationary effects and short data are avoided, and high recognition precision is achieved.

Description

Technical Field

[0001] This invention belongs to the field of bridge structural health monitoring technology, specifically relating to an automatic extraction method for the fundamental frequency of steel high-speed railway bridges under operational conditions. Background Technology

[0002] Steel-structured high-speed railway bridges are characterized by large spans, high stiffness, and heavy weight. During long-term operation, monitoring changes in their fundamental frequency (first-order natural frequency) is a crucial method for assessing the overall stiffness and structural health of the bridge. However, in actual operating environments, accurate extraction of the fundamental frequency faces significant challenges: 1. High stiffness and low vibration: Steel bridges have extremely high stiffness. Under random environmental excitation (such as wind and ground vibration), the structural vibration response is weak and the signal-to-noise ratio is extremely low. It is difficult to identify effective modes using conventional methods.

[0003] Second, the excitation source is complex: it is usually necessary to use the strong excitation (live load) generated by the train crossing the bridge to induce structural vibration. However, the train crossing the bridge introduces a huge additional mass (vehicle-bridge coupling effect) and non-stationary impact, which means that the identified frequency is often the "vehicle-bridge coupling system frequency" rather than the "structural fundamental frequency" of the bridge itself. In addition, the data is highly non-stationary, and traditional FFT analysis is prone to spectral ambiguity.

[0004] 3. Insufficient frequency resolution: Due to the short train crossing time and limited effective data length, traditional spectral analysis methods have low frequency resolution and are difficult to distinguish dense modal frequencies. Summary of the Invention

[0005] The purpose of this invention is to provide a method for extracting the inherent fundamental frequency of a bridge structure with high precision by using train live load as an effective excitation under operating conditions, while eliminating non-stationary interferences.

[0006] The technical solution of the present invention is as follows: 1. An automatic extraction method for the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions, characterized by comprising the following steps: [1] Sensor deployment and response signal acquisition Accelerometers were deployed at key locations on the bridge under test to continuously collect acceleration responses during operation and process the raw displacement signals from the sensors. [2] Operating condition identification and quasi-constant load data segmentation The energy envelope of sensor signals is used to identify the train crossing the bridge status, obtain the moment when the train is fully loaded on the bridge and running at a stable speed, and extract quasi-constant load data segments with good stability from the original displacement signals as valid signals; [3] Coherent weighted superposition of multiple measurement points The effective signals from multiple measurement points are weighted and superimposed using coherence as the weight to obtain a weighted superimposed signal. [4] Coarse frequency locking Perform FFT transformation on the weighted superimposed signal to determine the selected interval of the fundamental frequency; [5] Fine-grained base frequency identification Within the candidate interval of the fundamental frequency, a data covariance matrix is ​​constructed, eigenvalue decomposition is performed, and the MUSIC algorithm is applied to construct a pseudospectral function. The frequency corresponding to the pseudospectral peak is searched to obtain the fundamental frequency value. 【6】Statistical fusion output: Statistical analysis is performed on the identification results of multiple vehicles to remove outliers affected by random factors, and the fundamental frequency result with statistical stability is output.

[0007] This invention proposes an automatic method for extracting the fundamental frequency of steel high-speed railway bridges under operational conditions. It involves conducting artificial vibration tests without interrupting train operations, and the extracted data relies entirely on operational train data for automated monitoring, making it widely applicable. During signal extraction, coherent weighted superposition technology effectively solves the challenge of extracting signals from steel bridges with low vibration and high on-site noise, demonstrating strong anti-interference capabilities. Combining quasi-constant load segment screening and high-resolution spectrum estimation, it utilizes the strong excitation energy of live loads while avoiding frequency errors caused by non-stationary effects and short data points, resulting in high recognition accuracy. Attached Figure Description

[0008] Figure 1 This is a flowchart of the method of the present invention; Figure 2 Raw data collected from multiple accelerometers; Figure 3 The results of the calculation for working condition identification and quasi-dead load segmentation; Figure 4 The results are obtained by weighted superposition of multiple measurement points for the quasi-permanent load segment; Figure 5 For the FFT, perform a coarse frequency locking result; Figure 6 The result of fine-grained fundamental frequency identification for the MUSIC algorithm. Detailed Implementation

[0009] The following is combined Figure 1 The specific algorithm flow will be further described in detail for this invention.

[0010] Step 1: Multi-point acceleration response acquisition 1.1 Sensor Deployment Scheme Deployment at key locations on steel high-speed railway bridges An accelerometer, typically It is recommended to deploy sensors at key locations such as mid-span, L / 4, and 3L / 4 of the main beam to ensure the capture of the structure's first-order vibration mode characteristics. MEMS accelerometers should be used, with a range covering the vibration amplitude range (±2g) of the bridge under train loads. Deployment locations should prioritize areas with large structural response amplitudes and high signal-to-noise ratios, avoiding nodal points (where the vibration mode is zero). Simultaneously, the original displacement signal from the sensor is obtained by performing a second integration on the acceleration signal, such as... Figure 2 As shown.

[0011] 1.2 Sampling Frequency Setting According to the Nyquist theorem It should be 5-10 times greater than the estimated fundamental frequency, that is ; in Estimate the fundamental frequency for the bridge (usually 1-5 Hz). This refers to the sampling frequency. For steel-structured high-speed railway bridges, it is recommended... Hz, typically 100 Hz or higher.

[0012] Step 2: Perform load condition identification and extract quasi-dead load segments. 2.1 Energy Calculation by Sliding Window The energy envelope of sensor signals is used to identify the train's bridge-crossing status. The energy within a sliding time window is defined. : ; :time The sliding window energy value, in units of (m / s²)², reflects the vibration intensity of the signal near that moment; Discrete-time index. The number of sampling points within the window. : Index of sampling points within the window. Reference measuring point at time The signal value.

[0013] 2.2 Train Entry Determination when It was determined at that time that a train was crossing the bridge. This is the energy threshold for trains entering the bridge, expressed in (m / s²)². This threshold is typically set based on the background noise energy level when there are no trains, and can be taken as 3-5 times the background noise energy. ,in This represents the average energy level under no-vehicle conditions.

[0014] 2.3 Signal Variance Calculation Further calculation of the in-window signal variance Used to evaluate the stationarity of a signal: ; :time The sliding window variance, expressed in (m / s²)², reflects the degree of signal fluctuation near that moment. : Number of sampling points within the window; Discrete-time index; : Index of sampling points within the window. Reference measuring point at time The signal value; The average signal value within the window. .

[0015] 2.4 Calculation of Variance Volatility Calculate the volatility of the variance to assess the stability of the signal: ; :time The variance volatility is dimensionless and reflects the relative degree of fluctuation of the variance.

[0016] : within the time window Internal variance standard deviation To evaluate the window length (usually 10-20 seconds). : within the time window Internal variance The average value.

[0017] 2.5 Quasi-dead load segment determination If the following conditions are met simultaneously, the time period is determined to be a quasi-permanent load time period: 1) Energy conditions: (The train is on the bridge.) 2) Stability conditions: ,in To preset the stability threshold, it is usually taken as... . 3) Duration condition: Duration of quasi-dead load state ,in To meet the minimum duration requirement, it is usually taken as Seconds are used to ensure sufficient data length for frequency analysis.

[0018] like Figure 3As shown, vibration response data corresponding to a quasi-constant load time period that meets the conditions is extracted as the analysis segment. Within this segment, the train can be regarded as a moving constant load, and the excited vibration mainly reflects the inherent characteristics of the structure, avoiding the strong non-steady impact when the train enters and exits the bridge.

[0019] Step 3: Perform coherent weighted superposition of multi-point constant load segment data. 3.1 Selection of Reference Measurement Points The measurement point with the largest response energy in the effective signal is selected as the reference measurement point. ,in: Reference measurement point number, . Reference measuring point at time The effective signal value.

[0020] The selection criteria for reference measuring points are as follows: ; Reference measurement point number, . : Duration of the quasi-constant load segment. : The parameter that maximizes the objective function. Measurement point number, 3.2 Power spectral density calculation Calculate each measuring point With reference measuring point The cross power spectral density and the self power spectral density.

[0021] : No. The self-power spectral density at each measurement point is expressed in units of (m / s²)² / Hz, representing the signal at that measurement point at a frequency of [frequency missing]. Power at the location. Reference measuring point The self-power spectral density. Reference measuring point With the The cross-power spectral density between measurement points, expressed in (m / s²)² / Hz, represents the cross-power spectral density between two signals at different frequencies. Correlation at the location.

[0022] Power spectral density can be calculated using the Welch method or the periodogram method: ; in: : Fourier transform, . : Fourier transform. : . Expectation operator. Imaginary unit, .

[0023] 3.3 Calculation of coherence function Calculate the coherence function : ; in: :frequency The coherence function value (square coherence function) at point is dimensionless and its range is . . Indicates that the two signals are at the same frequency. The correlation is perfect (ideal case).

[0024] Indicates that the two signals are at the same frequency. They are completely unrelated.

[0025] : The magnitude (amplitude) of the cross-power spectral density. : The square of the cross-power spectral density mode. : No. The self-power spectral density at each measurement point.

[0026] : Self-power spectral density at the reference measurement point.

[0027] The coherence function reflects the degree of linear correlation between two signals at a specific frequency. For the fundamental frequency mode of the structure, the response at each measurement point should have high coherence; while for local noise and interference, the coherence between measurement points is low.

[0028] 3.4 Calculation of Average Coherence Coefficient Calculate the average coherence coefficient within the target frequency band. : ; in: The average coherence coefficient within the target frequency band is dimensionless and its value ranges from [value missing]. . The lower limit frequency of the target frequency band, in Hz, is usually taken as... ,in To estimate the fundamental frequency. The upper limit frequency of the target frequency band, in Hz, is usually taken as... .

[0029] Discretization implementation (frequency resolution is) ): ; in: Number of frequency points within the target frequency band . Frequency index . Frequency resolution, ,in The number of FFT points.

[0030] 3.5 Measurement Point Selection like Then the measuring point is included in the superposition set. ,in: The coherence coefficient threshold is usually taken as... This is used to filter out measurement points that are highly correlated with the reference measurement point. The set of measurement points participating in the overlay. ,and .

[0031] 3.6 Weight Calculation Calculate normalized weights : ; in: : No. The normalized weights for each measurement point are dimensionless and range from [value range missing]. And satisfy . Superimposed Sets The measurement point index in the system. The sum of the squares of the average coherence coefficients of all participating measurement points is used for normalization.

[0032] Weight Proportional to the average coherence coefficient, measurement points with high coherence (more likely to contain the structural fundamental mode) receive a greater weight, while measurement points with low coherence (potentially containing more noise) receive a smaller weight.

[0033] 3.7 Construction of Weighted Superposition Signals Constructing an equivalent single-channel response signal : ; The equivalent single-channel response signal after weighted superposition, in m / s². Sum the values ​​from all measurement points involved in the overlay. For example... Figure 4 As shown, this step enables the common structural modes (fundamental frequencies) at all measurement points to be superimposed and enhanced in phase, while uncorrelated noises cancel each other out, thereby significantly improving the signal-to-noise ratio.

[0034] Step 4: Use FFT for coarse frequency locking For superimposed signals Perform a Fast Fourier Transform (FFT) to obtain the power spectral density: ; in: :frequency The power spectral density value at that location is expressed in units of (m / s²)² / Hz.

[0035] : The Discrete Fourier Transform (DFT). . Number of signal sampling points. : The magnitude (amplitude).

[0036] like Figure 5 As shown, based on the design data and the peak position of the power spectrum, spectral peaks are searched near the estimated fundamental frequency (e.g., 0.5Hz-10Hz) to determine the candidate fundamental frequency range. , This is the lower limit frequency of the candidate frequency band. This represents the upper limit frequency of the candidate frequency band. The candidate interval should include the main peaks in the power spectrum, typically taken as a fraction of the peak frequency. scope.

[0037] Step 5: Use the MUSIC algorithm for fine-grained fundamental frequency identification. 5.1 Construction of Data Covariance Matrix Within the candidate interval, construct the data covariance matrix. First, the signal... Segmented processing, constructing a data matrix: ; : The data matrix The number of rows (usually taken as) ), The number of columns (usually taken as) or ). : The number of rows in the matrix, indicating the order of the model in the subspace analysis. : Number of columns in the matrix, indicating the length of the data used for estimation.

[0038] Calculate the covariance matrix of the data: ; : The covariance matrix. : The conjugate transpose of .

[0039] 5.2 On the covariance matrix Perform eigenvalue decomposition: ; : The eigenvector matrix has column vectors of . eigenvectors. : A diagonal matrix with diagonal elements of The eigenvalues ​​are arranged in descending order. : The conjugate transpose of .

[0040] 5.3 Noise Subspace Extraction Based on the magnitude of the eigenvalues, the eigenvector is divided into a signal subspace and a noise subspace: signal subspace : Feature vectors corresponding to larger feature values, with dimension . , Noise subspace : The feature vector corresponding to the smaller feature value, with dimension . .in The number of signal sources (for fundamental frequency extraction, it is usually taken as...). ). 5.4 Construction of MUSIC Pseudospectral Functions Constructing pseudospectral functions using the MUSIC algorithm: ; in: :frequency The MUSIC pseudospectral value at that frequency is dimensionless (or can be understood as relative power). The larger the value, the greater the probability that there is a signal component at that frequency. : The steering vector is defined as follows: ; Sampling interval The unit is seconds. Complex exponent term, representing frequency. At any moment The phase. : Transpose operator. : The conjugate transpose (Hermitian transpose) is... The row vector. : The conjugate transpose of the matrix, with dimension . . : The projection matrix of the noise subspace.

[0041] The basic principle of the MUSIC algorithm is: if the frequency... If it is the signal frequency, then the steering vector It should be located in the signal subspace and orthogonal to the noise subspace, that is... ,thereby A peak will occur.

[0042] 5.5 Fine-grained identification of fundamental frequency In candidate frequency band Within, a high-frequency resolution search (typically 0.01 Hz or less) is performed. The peak value, and the frequency corresponding to the peak value, is the fundamental frequency for fine identification. : ; in: : The estimated fundamental frequency obtained from identification, in Hz. : The parameter that maximizes the objective function.

[0043] like Figure 6 As shown, the MUSIC algorithm overcomes the frequency resolution limitation of short-time data (the resolution of traditional FFT is...). It can achieve super-resolution frequency estimation with a frequency resolution of 0.01 Hz or higher, which is more than 10 times better than FFT.

[0044] Step 6: Baseband Statistical Fusion Output 6.1 Data Collection for Multiple Train Routes The frequency of a single train crossing a bridge may be affected by accidental factors (such as wheel-rail wear, uneven cargo loading, track irregularities, etc.), leading to fluctuations in the fundamental frequency estimate. This method collects... The identification results of the train crossing the bridge form a set of fundamental frequency estimates: ; in: : Number of times trains crossed bridges collected (number of analysis segments) : No. The train crosses the bridge (the first) The fundamental frequency estimate obtained by identifying (each analysis segment) The unit is Hz.

[0045] 6.2 Outlier Removal Outliers (abnormal values) are removed using statistical methods. The number of remaining valid estimates after outlier removal is denoted as [missing information]. ,in .

[0046] 6.3 Calculation of Statistical Mean Calculate the mean of the remaining effective estimates as the final fundamental frequency: ; in: : The statistical average value of the fundamental frequency (final extracted result), in Hz. : The number of valid estimates remaining after removing outliers. : Sum of all valid estimates.

[0047] This invention proposes an automatic extraction method for the fundamental frequency of steel high-speed railway bridges under operating conditions. The method involves conducting artificial vibration tests without interrupting train operations, and the extracted data relies entirely on operating train data for automated monitoring. It has strong anti-interference capabilities, high recognition accuracy, and wide applicability.

Claims

1. An automatic method for extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions, characterized in that, Includes the following steps: [1] Sensor deployment and response signal acquisition Accelerometers were deployed at key locations on the bridge under test to continuously collect acceleration responses during operation and process the raw displacement signals from the sensors. [2] Operating condition identification and quasi-constant load data segmentation The energy envelope of sensor signals is used to identify the train crossing the bridge status, obtain the moment when the train is fully loaded on the bridge and running at a stable speed, and extract quasi-constant load data segments with good stability from the original displacement signals as valid signals; [3] Coherent weighted superposition of multiple measurement points The effective signals from multiple measurement points are weighted and superimposed using coherence as the weight to obtain a weighted superimposed signal. [4] Coarse frequency locking Perform FFT transformation on the weighted superimposed signal to determine the selected interval of the fundamental frequency; [5] Fine-grained base frequency identification Within the candidate interval of the fundamental frequency, a data covariance matrix is ​​constructed, eigenvalue decomposition is performed, and the MUSIC algorithm is applied to construct a pseudospectral function. The frequency corresponding to the pseudospectral peak is searched to obtain the fundamental frequency value. 【6】Statistical fusion output: Statistical analysis is performed on the identification results of multiple vehicles to remove outliers affected by random factors, and the fundamental frequency result with statistical stability is output.

2. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that: In step [1], N acceleration sensors are installed at the mid-span, 3L / 4 and L / 4 positions of the main beam of the steel high-speed railway bridge, where L is the length of the bridge.

3. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that: In step [1], the sampling frequency is 5-10 times the estimated bridge fundamental frequency.

4. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that, The identification steps for the quasi-permanent load segment in step [2] are as follows: 【2.1】 Sliding Window Energy Calculation The energy envelope of sensor signals is used to identify the train's crossing status, and the energy within a sliding time window is defined. ; :time The sliding window energy value reflects the vibration intensity of the signal near that moment; Discrete-time index; The number of sampling points within the window. : Index of sampling points within the window Reference measuring point at time The original displacement signal value; Reference measurement point number, ; 【2.2】 Train Entry Determination when It was determined at that time that a train was crossing the bridge. This is the energy threshold for trains entering the bridge. This threshold is usually set based on the background noise energy level when there are no trains, and can be taken as 3-5 times the background noise energy. ,in The average energy under no-vehicle conditions; 【2.3】 Calculation of signal variance Further calculation of the in-window signal variance Used to evaluate the stationarity of a signal: ； :time The variance of the sliding window reflects the degree of signal fluctuation near that moment; : Number of sampling points within the window; Discrete-time index; : Index of sampling points within the window; The mean of the original signal within the window. ; 【2.4】Calculation of Variance Volatility Calculate the volatility of the variance to assess the stability of the signal: ； :time The variance volatility; : within the time window Internal variance standard deviation To evaluate the window length, it is usually set to 10-20 seconds; : within the time window Internal variance The average value; 【2.5】 Quasi-dead load segment determination If the following conditions are met simultaneously, the time period is determined to be a quasi-permanent load time period: 1) Energy conditions: This indicates that the train is on the bridge; 2) Stability conditions: ,in To preset the stability threshold, it is usually taken as... ; 3) Duration condition: Duration of quasi-dead load state ,in To meet the minimum duration requirement, it is usually taken as seconds, to ensure sufficient data length for frequency analysis; The original signal data corresponding to the quasi-constant load time period that meets the conditions is extracted as the valid signal.

5. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that, Step [3] The multi-point coherent weighted superposition step is as follows: 【3.1】Selection of Reference Measurement Points The measurement point with the largest response energy was selected as the reference measurement point. , Reference measuring point at time The effective signal value; The selection criteria for reference measuring points are as follows: ； : Duration of the quasi-permanent load segment; The parameter that maximizes the objective function; Measurement point number, ; 【3.2】 Power spectral density calculation Calculate each measuring point With reference measuring point The cross-power spectral density and the self-power spectral density; : No. The self-power spectral density at each measurement point; Reference measuring point The self-power spectral density; Reference measuring point With the Cross-power spectral density between measurement points; Power spectral density can be calculated using the Welch method or the periodogram method: ； : Fourier transform, ; : Fourier transform; : The complex conjugate; Expectation operator; Imaginary unit, ; 【3.3】Calculation of coherence function Calculate the coherence function : ； :frequency The coherence function value at point [location] takes values ​​within the range of [range]. ; : The modulus of the cross-power spectral density; : No. The self-power spectral density at each measurement point; : Self-power spectral density at the reference measurement point; 【3.4】Calculation of average coherence coefficient Calculate the average coherence coefficient within the target frequency band. : ； The average coherence coefficient within the target frequency band, with a value range of [value missing]. ; The lower limit frequency of the target frequency band, in Hz, is usually taken as... ,in To estimate the fundamental frequency; The upper limit frequency of the target frequency band, in Hz, is usually taken as... ; Discretization implementation (frequency resolution is) ): ； Number of frequency points within the target frequency band ; Frequency index ; Frequency resolution, ,in The number of FFT points; 【3.5】Surveying Point Selection like Then the measuring point is included in the superposition set. , The coherence coefficient threshold is usually taken as... This is used to filter out measurement points that are related to the height of the reference measurement point. The set of measurement points participating in the overlay. ,and ; 【3.6】Weight Calculation ； : No. The normalized weights for each measurement point, with values ​​ranging from [value range missing]. And satisfy ; Superimposed Sets The measurement point index in the middle; The sum of the squares of the average coherence coefficients of all participating measurement points is used for normalization. 【3.7】Construction of weighted superposition signals ； The equivalent single-channel response signal after weighted superposition; : Sum all measurement points involved in the superposition.

6. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that, Step [4] The coarse frequency locking step is as follows: For superimposed signals Perform a Fast Fourier Transform (FFT) to obtain the power spectral density: ； :frequency The power spectral density value at that location; : The Discrete Fourier Transform (DFT). ; Number of signal sampling points; : The magnitude (amplitude); based on design data and the peak position of the power spectrum, search for spectral peaks near the estimated fundamental frequency to determine the candidate fundamental frequency range. , This is the lower limit frequency of the candidate frequency band. This represents the upper limit frequency of the candidate frequency band; The candidate interval should include the main peaks in the power spectrum, typically taking the peak frequency. scope.

7. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that, Step [5] The fine identification of the fundamental frequency is as follows: 【5.1】Construction of Data Covariance Matrix Signal Segmented processing, constructing a data matrix: ； : The data matrix The number of rows is usually taken as... ; For the number of columns, usually take or ; Calculate the covariance matrix of the data: ； : The covariance matrix; : The conjugate transpose of ; 【5.2】On the covariance matrix Perform eigenvalue decomposition: ； : The eigenvector matrix has column vectors of . eigenvectors; : A diagonal matrix with diagonal elements of The eigenvalues ​​are arranged in descending order; : The conjugate transpose of ; 【5.3】Noise Subspace Extraction Based on the magnitude of the eigenvalues, the eigenvector is divided into a signal subspace and a noise subspace: signal subspace : Feature vectors corresponding to larger feature values, with dimension . Noise subspace : Feature vectors corresponding to smaller feature values, with dimension ,in The number of signal sources is usually taken as... ; 【5.4】Construction of MUSIC pseudospectral function Constructing pseudospectral functions using the MUSIC algorithm: ； :frequency MUSIC pseudospectral value at the location; : The steering vector is defined as follows: ； Sampling interval ; Complex exponent term, representing frequency. At any moment The phase; : Transpose operator; : The conjugate transpose of is The row vector; : The conjugate transpose matrix with dimension ; : The projection matrix of the noise subspace; 【5.5】Fine-frequency identification In candidate frequency band Within, a high-frequency resolution search is performed, typically at 0.01 Hz. The peak value, and the frequency corresponding to the peak value, is the fundamental frequency for fine identification. : ； The estimated fundamental frequency obtained from identification, in Hz. : The parameter that maximizes the objective function.

8. The method for automatically extracting the fundamental frequency of a steel-structured high-speed railway bridge under operating conditions according to claim 1, characterized in that, Step [6] The fundamental frequency statistical fusion output steps are as follows: 【6.1】Data collection from multiple train trips collect The identification results of the train crossing the bridge form a set of fundamental frequency estimates: ； The number of times trains crossed bridges in the collected valid data; : No. The fundamental frequency estimate obtained from the identification of the train crossing the bridge. The unit is Hz; 【6.2】Outlier Removal Outliers were removed using statistical methods. After outlier removal, the number of remaining valid estimates was denoted as . ,in ; 【6.3】Calculation of Statistical Mean Calculate the mean of the remaining effective estimates as the final fundamental frequency: ； : The statistical average value of the fundamental frequency; : The number of valid estimates remaining after removing outliers; : Sum of all valid estimates.

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