Bridge operational modal analysis method based on fusion of structural dynamics and gaussian process

Abstract

The application discloses a bridge operation modal analysis method based on structural dynamics and Gaussian process fusion, and can be used for bridge structure health monitoring. Acceleration sensors are arranged at key positions of the bridge to collect vibration responses in an operation state; mode force power spectral densities of each mode are extracted through frequency domain analysis as external excitation strength indexes. The indexes are taken as Gaussian process inputs and hidden variables are introduced to quantize influences of excitation changes on natural frequencies, damping ratios and vibration modes; an expectation-maximization algorithm is used to iteratively identify structural dynamics parameters and Gaussian process hyperparameters. Finally, the bridge modal parameters after excitation disturbances such as wind and vehicle flow are removed and the hyperparameters representing working conditions are obtained, and long-term tracking, evaluation and prediction of the bridge structure health are realized.

Description

Technical Field

[0001] This invention relates to the field of bridge structural health monitoring and dynamic analysis technology, and in particular to a bridge operation modal analysis method based on the fusion of structural dynamics and Gaussian processes, which is used to identify bridge dynamic modal parameters and quantify external excitation effects under bridge operation conditions. Background Technology

[0002] As a critical component of transportation infrastructure, the safety and reliability of bridge structures directly impact public safety and economic development. Bridge structural health monitoring typically involves long-term, continuous acquisition of dynamic response data (e.g., acceleration response) under operational conditions to assess damage and degradation. Among numerous methods, bridge operational modal analysis is widely used in bridge structural health monitoring to estimate modal parameters, including natural frequencies, damping ratios, and mode shapes. These modal parameters characterize the dynamic properties of bridges under actual operating conditions, providing crucial information for bridge condition assessment and diagnosis.

[0003] However, bridges are subjected to external excitations such as traffic loads and wind loads during actual operation, causing deviations in the identified bridge modal parameters. For example, during peak traffic hours, the frequent passage of large, heavy vehicles causes a momentary increase in structural stiffness, resulting in a short-term increase in the frequencies identified through operational modal analysis. When wind speeds increase from light to strong, the structure experiences more severe vortex-induced vibrations, causing an upward shift in the identified natural frequencies, which in turn leads to deviations in modal parameter estimates from their true values. These environmental factors and traffic load interferences often prevent monitoring data from accurately reflecting the structural health status of the structure itself, reducing the accuracy and reliability of health assessments and damage identification.

[0004] Currently, common operational modal analysis methods include three types of modal identification techniques: time domain, frequency domain, and time-frequency domain. However, changes in operating conditions lead to shifts in the modal parameters identified under different operating conditions, increasing the uncertainty of parameter estimation in traditional operational modal analysis methods and causing misjudgments or false alarms during long-term monitoring. To address parameter drift caused by changes in operating conditions, existing research has attempted to use data-driven models to characterize the mapping relationship between changes in operating conditions and modal parameters. However, purely data-driven models usually lack clear physical interpretability and are difficult to effectively distinguish between external excitation effects and structural damage effects. Therefore, there is an urgent need for a fusion method that can simultaneously identify bridge modal parameters and systematically correct external excitation effects under bridge operating conditions. This method should maintain the flexibility of data-driven models while possessing physical meaning, effectively distinguishing parameter variations caused by external excitations and structural damage, thereby improving the accuracy and robustness of damage identification in long-term bridge structural health monitoring. Summary of the Invention

[0005] The purpose of this invention is to provide a bridge operation modal analysis method based on the fusion of structural dynamics and Gaussian processes. This method combines a bridge structural dynamics model with a Gaussian process model, and quantifies the impact of external excitations (such as wind loads and traffic loads) on bridge modal parameters by introducing latent variables. It systematically corrects the identification bias caused by changes in external excitations in traditional bridge operation modal identification methods, thereby extracting dynamic characteristics that better reflect the structural state of the bridge and improving the accuracy and reliability of bridge structural health assessment.

[0006] This invention provides a bridge operation modal analysis method based on the fusion of structural dynamics and Gaussian processes. Based on the bridge's measured operational response data, the method uses the physical-data fusion model to calculate the modal force power spectral density, which is then used as an indirect indicator of the intensity of external excitation on the bridge. Simultaneously, a Gaussian process model is employed to model the effects of external excitation, eliminating the nonlinear influence of external excitation changes on the bridge's dynamic modal parameters. Convergence is then achieved through iterative updates, ultimately obtaining the bridge modal parameters after eliminating the influence of external excitation and the Gaussian process model parameters characterizing the impact of bridge operation conditions. These parameters are used for bridge structural health assessment and dynamic analysis. The method specifically includes the following steps:

[0007] Step 1: Install acceleration sensors on the bridge structure to collect acceleration time history data of the bridge under operating conditions, and segment the acceleration time history data in chronological order using a sliding window.

[0008] Step 2: Perform frequency domain transformation on each segment of acceleration time history data to obtain the corresponding frequency domain response;

[0009] Step 3: Construct a bridge structural dynamics model that includes the effects of external excitations, wherein the effects of external excitations are characterized by an embedded Gaussian process model;

[0010] Step 4: Initialize the bridge structure dynamics model parameters, modal force power spectral density, and Gaussian process hyperparameters;

[0011] Step 5: Introduce the output of the Gaussian process model as a latent variable, and use variational inference to perform a Gaussian approximation on the posterior distribution of the latent variable;

[0012] Step 6: Iteratively update the latent variables, Gaussian process hyperparameters, bridge structural dynamics model parameters, and modal force power spectral density using the expectation-maximization algorithm;

[0013] Step 7: Output the bridge modal parameters after removing the influence of external excitation changes, as well as the Gaussian process hyperparameters characterizing the influence of bridge operating conditions, to achieve long-term tracking of bridge dynamic characteristics.

[0014] Furthermore, in step 1, the sliding window is a segment of acceleration signal, the window length is the duration of the signal segment, and the window step size is the length by which the window translates on the time axis.

[0015] Furthermore, step 2 specifically involves: the Fast Fourier Transform employs a one-sided power spectral density, let { } for in The acceleration time history response of the structure on each degree of freedom is measured. express 3D real vector space, For each data channel, the Fast Fourier Transform scaled on one side is defined as:

[0016]

[0017] The corresponding frequency is:

[0018]

[0019] in, The imaginary unit satisfies ; The sampling time interval satisfies the following condition with the sampling frequency: ; The time sampling point number indicates the number of the sampling points within the window. A discrete sampling point; The frequency line number corresponds to the [number]th ... One spectral line; For the first The physical frequency corresponding to each spectral line.

[0020] Furthermore, step 3 specifically involves constructing a physical-data fusion model that incorporates the influence of external stimuli. The modal motion equation expression for each dataset segment is as follows:

[0021]

[0022] in, , , These are the fast Fourier transforms of acceleration, velocity, and displacement in the decoupled structural dynamic modal equations, respectively. Natural angular frequency (corresponding to natural frequency) ); The damping ratio; Modal force; The output of the Gaussian process model used to characterize the influence of changes in external excitation on the bridge is given, and let... Represents the sequence of incentive effect coefficients; This represents the number of data segments.

[0023] The Gaussian process model is defined as follows: Its mean function With covariance function They are respectively:

[0024]

[0025]

[0026] The elements of the covariance matrix are constructed using a squared exponential kernel function:

[0027]

[0028] in, The input variable for the Gaussian process model is the modal force power spectral density, which can indirectly characterize the intensity of external excitation. This represents another input point for the input variables of the Gaussian process model; The first , No. The input values ​​corresponding to the segment data; The notation for a Gaussian process indicates that it follows a functional distribution characterized by the mean and covariance functions; Represents a random function in a Gaussian process, mapping from input to output; The mathematical expectation operator represents the mean of a random quantity; The first element of the covariance matrix represents the first element of the covariance matrix. One element; For the calculation of norm; Represents the Kronecker function; This is the set of hyperparameters for the Gaussian process kernel function; The standard deviation of the signal. For the feature length scale, This represents the standard deviation of the noise.

[0029] Furthermore, in step 4, the incentive effect coefficient is... Initialize to zero; and calculate the modal force power spectral density using the following formula. .

[0030]

[0031] in, for Modal forces corresponding to the first mode; for The conjugate of complex numbers.

[0032] Furthermore, step 5 specifically involves: using variational inference to perform a Gaussian approximation on the posterior distribution of the latent variables; specifically, it involves: approximating the true posterior distribution... Approximate as mean vector With covariance matrix The Gaussian distribution, and the corresponding variational distribution are defined as follows: .

[0033] Calculate according to the following formula Gaussian approximate distribution :

[0034]

[0035]

[0036] in, Indicates to The variational approximation distribution of the posterior is approximated by a Gaussian distribution; This represents the set of bridge observation data after segmentation; For the first The response data corresponding to the segment; This represents the total number of data segments. This represents the set of parameters for the bridge structural dynamics model; Indicates the hyperparameters of a Gaussian process; Represents the covariance function of a Gaussian process The inverse matrix; The mean function representing a Gaussian process; Indicates transpose; This represents the inverse operation; This is an operator that forms a diagonal matrix from the elements within the parentheses.

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] in, This represents the objective function term used to update parameters in variational inference; Indicates the objective function for the th Partial derivatives of the mean of the latent variables; Indicates the objective function for the th Partial derivative of segment covariance; This indicates that the trace of the element within the parentheses has been calculated. Indicates the number of frequency points used in the calculation; Represents the frequency domain covariance matrix; For the first The observed frequency domain response vector at each frequency point; Represents the operation of conjugate complex numbers; An intermediate matrix for deriving and simplifying the formula; Frequency point The amplitude of the standard value at the location; This is the mode shape vector for this mode; To measure the noise power spectral density; for The identity matrix; For the dynamic parameters and The related terms of the complex frequency response determined jointly; Indicates the dimensionless frequency ratio; Indicates the first The angular frequencies corresponding to each spectral line; The frequency of the current frequency point; For the first The physical frequency corresponding to each spectral line; Indicates the damping ratio. This represents the modal force power spectral density.

[0046] Furthermore, step 6 specifically involves the expectation-maximization algorithm, which is as follows:

[0047] Step E: Calculate the expected value of the log-likelihood function.

[0048]

[0049] M-step: By maximizing To update the parameters.

[0050]

[0051] objective function Decomposed into two parts:

[0052]

[0053] in:

[0054]

[0055]

[0056] in, In the EM algorithm function; This indicates the part related to the consistency of the observed data; This represents the part related to the latent variable; This indicates taking the natural logarithm; The operator that takes the parameter that maximizes the objective function; Indicates the number of iterations; Represents the latent variable Given observation data With structural parameters Modal force power spectral density Gaussian process hyperparameters The t Mathematical expectation of the posterior distribution under the next iteration value; Indicates that under given structural parameters Modal force power spectral density and latent variables At that time, the observed data The likelihood function; Indicates that given input and Gaussian process hyperparameters hour Distribution; This indicates determinant operations.

[0057] Repeat the following three iterative steps: Calculate the incentive effect coefficient. By maximizing Updating Gaussian process hyperparameters By maximizing Update modal parameters and modal force power spectral density .

[0058] Furthermore, step 7 specifically involves: the final modal force power spectral density, the structural dynamics model parameters after eliminating the influence of external excitations, and the Gaussian process hyperparameters reflecting the influence of operating conditions. The convergence criterion used is as follows:

[0059]

[0060] in, This represents the objective function value calculated after the current iteration update during the EM iteration process; This represents the objective function value corresponding to the previous iteration in the EM iteration process; This represents the convergence tolerance, and it is set to... .

[0061] The present invention also discloses a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method of the present invention.

[0062] The present invention also discloses a computer-readable storage medium having a computer program / instructions stored thereon, which, when executed by a processor, implements the steps of the method of the present invention.

[0063] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0064] (1) A method was proposed that modal force power spectral density be introduced as an indirect proxy variable for excitation intensity, and further construct a nonlinear mapping relationship between excitation and modal parameters through a Gaussian process. This method breaks through the limitations of traditional techniques by ignoring situations where wind load, traffic load, etc., cannot be directly measured, and realizes the quantification and compensation of excitation effect under conditions where external excitation cannot be directly monitored, thereby improving the accuracy of modal parameter identification.

[0065] (2) This invention couples the structural dynamics physical model with the Gaussian process data-driven model through latent variables, which not only preserves the interpretability of the physical mechanism, but also utilizes the Gaussian process to flexibly capture the complex nonlinear relationship between excitation and response. This fusion framework not only suppresses the deviation caused by excitation, but also provides a correction mechanism with a clear physical basis for parameter identification, thereby achieving accurate identification of modal parameters.

[0066] (3) Existing methods typically handle modal parameter identification and data model training separately, resulting in low computational efficiency and neglecting parameter coupling effects. This invention proposes an iterative optimization framework based on the expectation-maximization algorithm, combined with variational inference techniques, to achieve simultaneous joint estimation of modal parameters and Gaussian process hyperparameters. This strategy reduces computational complexity, accelerates algorithm convergence, and improves the robustness of parameter estimation.

[0067] (4) Through dual verification using synthetic data and actual bridge monitoring data, it is demonstrated that the identification deviation of the natural frequency and damping ratio of this method is smaller than that of the traditional method when the excitation intensity changes. The key modal parameters identified by this invention remain stable under variable excitation, avoiding misjudging the excitation effect as structural damage and significantly reducing the false alarm rate of long-term monitoring.

[0068] (5) The modal parameters output by this invention, after eliminating external excitations, are not only used to extract damage-sensitive indicators to eliminate environmental and excitation interference and highlight the actual structural performance degradation, but also used for life prediction and maintenance planning, thereby providing strong data support for the management of the entire life cycle of the structure. Attached Figure Description

[0069] Figure 1 This is a time history diagram of the simulation data of the present invention;

[0070] Figure 2 This is a time history diagram of actual bridge cable acceleration monitoring data according to the present invention;

[0071] Figure 3 This is a power spectral density plot of the simulation data of the present invention;

[0072] Figure 4 This is a power spectral density diagram of actual bridge cable acceleration monitoring data according to the present invention;

[0073] Figure 5 This is a construction diagram of the fusion model of the present invention;

[0074] Figure 6 This is a three-dimensional visualization of the objective function of the present invention;

[0075] Figure 7 This is a flowchart of the structural dynamic characteristic identification method of the present invention;

[0076] Figure 8 This is a frequency offset relationship diagram based on the simulation data assumptions of the present invention;

[0077] Figure 9 This is an overall layout diagram of a bridge according to the present invention. Detailed Implementation

[0078] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0079] The bridge operation modal analysis method based on the fusion of structural dynamics and Gaussian processes of the present invention includes the following steps:

[0080] Step 1: Analyze the simulated data and long-term monitoring data of a suspension bridge, setting each segment to 10 minutes. Figure 1 The time history record of a representative data segment from the simulation data is shown. Figure 2 The time history record of a representative data segment of the actual bridge data is displayed.

[0081] Step 2: Perform a Fast Fourier Transform on each data segment. Figure 3 The power spectral density of the simulated data is shown, where a significant modal peak at approximately 3 Hz indicates the validity of the synthesized data. Figure 4 The power spectral density of the real bridge data is given, and two modes affected by external excitation are selected for subsequent analysis.

[0082] Step 3: For the classical damped structure, construct a physical-data fusion model that incorporates the influence of external excitations. The modal motion equation expression for each data segment is as follows:

[0083]

[0084] in, , , These are the fast Fourier transforms of acceleration, velocity, and displacement in the decoupled structural dynamic modal equations, respectively. Natural angular frequency (corresponding to natural frequency) ); The damping ratio; Modal force; The output of the Gaussian process model used to characterize the influence of changes in external excitation on the bridge is given, and let... Represents the sequence of incentive effect coefficients; This represents the number of data segments.

[0085] The Gaussian process model is defined as follows: Its mean function With covariance function They are respectively:

[0086]

[0087]

[0088] The elements of the covariance matrix are constructed using a squared exponential kernel function:

[0089]

[0090] in, The input variable for the Gaussian process model is the modal force power spectral density, which can indirectly characterize the intensity of external excitation. This represents another input point for the input variables of the Gaussian process model; The first , No. The input values ​​corresponding to the segment data; The notation for a Gaussian process indicates that it follows a functional distribution characterized by the mean and covariance functions; Represents a random function in a Gaussian process, mapping from input to output; The mathematical expectation operator represents the mean of a random quantity; The first element of the covariance matrix represents the first element of the covariance matrix. One element; For the calculation of norm; Represents the Kronecker function; This is the set of hyperparameters for the Gaussian process kernel function; The standard deviation of the signal. For the feature length scale, This represents the standard deviation of the noise. The overall model framework is as follows: Figure 5 As shown.

[0091] Step 4: Calculate the incentive effect coefficient. Initialize to zero; and calculate the modal force power spectral density using the following formula. .

[0092]

[0093] in, for Modal forces corresponding to the first mode; for The conjugate of complex numbers.

[0094] Step 5: Applying variational inference to the Gaussian approximation of the posterior distribution of the latent variables. Specifically, this involves approximating the true posterior distribution... Approximate as mean vector With covariance matrix The Gaussian distribution of is given by . The corresponding variational distribution is defined as . .

[0095] Calculate according to the following formula Gaussian approximate distribution :

[0096]

[0097]

[0098] in, Indicates to The variational approximation distribution of the posterior is approximated by a Gaussian distribution; This represents the set of bridge observation data after segmentation; For the first The response data corresponding to the segment; This represents the total number of data segments. This represents the set of parameters for the bridge structural dynamics model; Indicates the hyperparameters of a Gaussian process; Represents the covariance function of a Gaussian process The inverse matrix; The mean function representing a Gaussian process; Indicates transpose; This represents the inverse operation; This is an operator that forms a diagonal matrix from the elements within the parentheses.

[0099]

[0100]

[0101]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107] in, This represents the objective function term used to update parameters in variational inference; Indicates the objective function for the th Partial derivatives of the mean of the latent variables; Indicates the objective function for the th Partial derivative of segment covariance; This indicates that the trace of the element within the parentheses has been calculated. Indicates the number of frequency points used in the calculation; Represents the frequency domain covariance matrix; For the first The observed frequency domain response vector at each frequency point; Represents the operation of conjugate complex numbers; An intermediate matrix for deriving and simplifying the formula; Frequency point The amplitude of the standard value at the location; This is the mode shape vector for this mode; To measure the noise power spectral density; for The identity matrix; For the dynamic parameters and The related terms of the complex frequency response determined jointly; Indicates the dimensionless frequency ratio; Indicates the first The angular frequencies corresponding to each spectral line; The frequency of the current frequency point; For the first The physical frequency corresponding to each spectral line; This indicates the damping ratio.

[0108] Step 6: Iteratively update the parameters using the expectation-maximization algorithm.

[0109] Step E: Calculate the expected value of the log-likelihood function.

[0110]

[0111] M-step: By maximizing To update the parameters.

[0112]

[0113] objective function Decomposed into two parts:

[0114]

[0115] in, In the EM algorithm function; This indicates the part related to the consistency of the observed data; This represents the part related to the latent variable;

[0116] Repeat the following three iterative steps: Calculate the incentive effect coefficient. By maximizing Updating Gaussian process hyperparameters By maximizing Update modal parameters and modal force power spectral density .

[0117] Step 7: The convergence criteria used for the final modal force power spectral density, structural dynamics model parameters after eliminating the influence of external excitations, and Gaussian process hyperparameters reflecting the influence of operating conditions are as follows:

[0118]

[0119] in, This represents the objective function value calculated after the current iteration update during the EM iteration process; This represents the objective function value corresponding to the previous iteration in the EM iteration process; This represents the convergence tolerance, which is set to... The objective function value of the algorithm in the frequency and damping ratio space. And the convergence of the algorithm, such as Figure 6 As shown.

[0120] The flowchart of the bridge operation modal analysis method based on the fusion of structural dynamics and Gaussian processes is as follows: Figure 7 As shown. The frequency offset relationship set in the simulation data is as follows. Figure 8 As shown, the overall layout of a suspension bridge is as follows: Figure 9As shown in the figure. Based on the aforementioned process, the simulated data and the measured monitoring data of the suspension bridge over 15 consecutive days were calculated and analyzed. Tables 1 and 2 respectively show the comparison of the bridge modal parameter identification results of the method of the present invention (FM-OMA) and the traditional bridge operation modal analysis method (C-OMA). The results show that the method of the present invention can accurately and stably identify bridge modal parameters under the disturbance of external excitation changes (such as wind load and traffic load). Compared with the traditional bridge operation modal analysis method, the method of the present invention has smaller identification bias and higher result stability, and can be used for bridge structural health monitoring and dynamic analysis. In addition, the method of the present invention has broad engineering application prospects and can be used for bridge structural health assessment, life prediction, damage identification, and long-term maintenance planning.

[0121] Table 1. Comparison of bridge modal parameter identification results based on simulation data between the method of the present invention (FM-OMA) and the traditional bridge operation modal analysis method (C-OMA).

[0122]

[0123] Table 2. Comparison of bridge modal parameter identification results based on measured data of a suspension bridge using the method of the present invention (FM-OMA) and the traditional bridge operation modal analysis method (C-OMA).

[0124]

[0125] The above embodiments are merely preferred embodiments of the present invention. It should be noted that those skilled in the art can make several improvements and equivalent substitutions without departing from the principle of the present invention. All such improvements and equivalent substitutions to the claims of the present invention fall within the protection scope of the present invention.

Claims

1. A bridge operational modal analysis method based on fusion of structural dynamics and Gaussian process, characterized in that, The method comprises the following steps: Step 1, arranging acceleration sensors on the bridge structure to collect acceleration time history data of the bridge under operating conditions, and segmenting the acceleration time history data in time sequence by using a sliding window; Step 2, performing frequency domain transformation on each segment of the acceleration time history data to obtain a corresponding frequency domain response; Step 3, constructing a bridge structure dynamics model containing external excitation effects, wherein the external excitation effects are represented by an embedded Gaussian process model; Step 4, initializing the bridge structure dynamics model parameters, modal force power spectral density, and Gaussian process hyperparameters; Step 5, introducing the Gaussian process model output as a latent variable, and using variational inference to perform Gaussian approximation on the posterior distribution of the latent variable; Step 6, iteratively updating the latent variable, Gaussian process hyperparameters, bridge structure dynamics model parameters, and modal force power spectral density by using an expectation-maximization algorithm; Step 7, outputting the bridge modal parameters after removing the influence of external excitation changes and the Gaussian process hyperparameters representing the influence of bridge operating conditions, to realize long-term tracking of the bridge dynamics characteristics.

2. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes according to claim 1, characterized in that: In the step 1, the window is a time period of the bridge acceleration time history signal, the window length is the duration of the time period, and the window step is the translation length of adjacent windows on the time axis.

3. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 1, wherein: In step 2, the bridge acceleration time-history signal in the sliding window is subjected to fast Fourier transform and the amplitude scaling method of one-sided spectrum is adopted; let } be the acceleration time-history response of the structure on measuring degrees of freedom, represent dimensional real vector space, be the sample point number of each data channel, and the fast Fourier transform on one side scaling is defined as: ； The corresponding frequency is: ； wherein, is the imaginary unit, satisfying ; is the sampling time interval, satisfying ; is the time sample point number, representing the th discrete sample point in the window; is the frequency line number, corresponding to the th spectral line in the discrete frequency domain; is the physical frequency corresponding to the th spectral line.

4. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 1, wherein: In step 3, the physical-data fusion model incorporating the influence of the change of the external excitation of the bridge is constructed, and the modal equation of motion of each order mode after decoupling is: The modal equation of motion after decoupling of the second order mode is: ； wherein, , , are the fast Fourier transform of the acceleration, velocity and displacement of the decoupled structural dynamic modal equation respectively; is the natural angular frequency, corresponding to the natural frequency ; is the damping ratio; is the modal force; is the Gaussian process model output for characterizing the effect of the variation of the external excitation of the bridge, and let denote the sequence of excitation effect coefficients; is the number of data segments; The Gaussian process model is defined as ; its mean function and covariance function respectively are: ； ； And a square exponential kernel function is used to construct the covariance matrix elements: ； in, The input variable for the Gaussian process model is the modal force power spectral density, which can indirectly characterize the intensity of external excitation. This represents another input point for the input variables of the Gaussian process model; The first , No. The input values ​​corresponding to the segment data; The notation for a Gaussian process indicates that it follows a functional distribution characterized by the mean and covariance functions; Represents a random function in a Gaussian process, mapping from input to output; The mathematical expectation operator represents the mean of a random quantity; The first element of the covariance matrix represents the first element of the covariance matrix. One element; For the calculation of norm; Represents the Kronecker function; This is the set of hyperparameters for the Gaussian process kernel function; The standard deviation of the signal. For the feature length scale, This represents the standard deviation of the noise.

5. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 1, wherein: In step 4, the excitation effect coefficient is initialized to zero; and the modal force power spectral density is calculated from ; ； wherein is the modal force corresponding to the mode of the is the conjugate complex of 6. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 1, wherein: In step 5, the posterior distribution of the latent variables is approximated as a Gaussian using variational inference, specifically: the true posterior distribution is approximated as a Gaussian with mean vector and covariance matrix , and the corresponding variational distribution is defined as ; where denotes the Gaussian process hyperparameters; denotes the modal parameters, denotes the modal force power spectral density, and the Gaussian approximation distribution is computed as .

7. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 6, wherein: In the step 6, the expectation-maximization algorithm is used to update the parameters, which specifically includes: E step: calculating the expected value of the log-likelihood function; ； M-step: Update parameters by maximizing ; ； Objective function Breaks down into two parts: ； wherein, represents the EM algorithm in ; represents the part related to the consistency with the observed data; represents the part related to the latent variables; By repeating the following three steps iteratively: Compute excitation effect coefficient ; by maximizing Update Gaussian process hyperparameters ; by maximizing Update modal parameters and modal force power spectral density .

8. The bridge operational modal analysis method based on fusion of structural dynamics and Gaussian processes of claim 1, wherein: In the step 7, the following convergence criterion is used for the final obtained modal force power spectral density, bridge structure dynamics model parameters after removing the influence of bridge external excitation, and Gaussian process hyperparameters reflecting the influence of bridge operating conditions: ； wherein, represents the objective function value calculated after the current iteration update in the EM iteration process; represents the objective function value corresponding to the previous iteration in the EM iteration process; represents the convergence tolerance.

9. A computer apparatus comprising a memory, a processor, and a computer program stored on the memory, wherein the computer program, when executed by the processor, causes the processor to perform the method of any one of claims 1 to 8. The processor executes the computer program to realize the steps of the method of claim 1.

10. A computer readable storage medium having stored thereon computer programs / instructions, characterized in that, The computer program / instructions are executed by the processor to realize the steps of the method of claim 1.

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