A multi-field coupling cable force bayesian prediction method of a cable-damper system

Abstract

This invention discloses a multi-field coupled Bayesian prediction method for cable-damper systems, belonging to the field of bridge structural health monitoring technology. The method first establishes a dynamic stiffness analysis model of the cable-damper system considering sag, bending stiffness, tilt angle, and damper parameters, and solves for the theoretical modal frequencies through extreme value optimization. Second, using cable force, cable section moment of inertia, and damping coefficient as parameters to be identified, a three-parameter joint identification is performed based on minimizing the weighted frequency error to obtain high-precision benchmark parameters. Finally, using the benchmark cable force as a reference, an incremental regression model of cable force under multi-source environmental inputs such as traffic load, temperature, and wind is established. A normal-inverse gamma conjugate prior is used for Bayesian updating, and the posterior prediction distribution is analytically solved to output the mean and probability prediction interval of the cable force prediction. This invention effectively solves the problem of quantifying the uncertainty of cable force prediction under the influence of parameter coupling and environmental effects.

Description

Technical Field

[0001] This invention relates to the field of bridge structural health monitoring and cable parameter identification technology, specifically to a cable-damper system cable force Bayesian update and probabilistic prediction method. Background Technology

[0002] Cable-stayed bridges are one of the main types of long-span bridges. As key load-bearing components, the cable stress state directly determines the bridge structure's alignment, internal force distribution, and service safety. During their service life, cables are subjected to multiple environmental factors such as traffic loads, temperature changes, and wind loads, resulting in significant time-varying cable stress. Accurately obtaining the cable stress state and its changing patterns is the core foundation for bridge health monitoring and condition assessment.

[0003] Existing cable parameter identification methods mostly focus on deterministic cable force identification under specific working conditions, aiming to minimize the error between the measured frequency and the theoretical frequency. For cable systems with external dampers, there is a strong coupling relationship between cable force, moment of inertia, and damper damping coefficient. When cable force is used as the sole identification parameter, the deviation between the moment of inertia and the damping parameter will directly translate into cable force identification error, causing the evaluation results to deviate from the true state.

[0004] At the same time, existing cable force Bayesian prediction methods are not fully integrated with the accurate dynamic model of the cable-damper system and the results of benchmark parameter identification. It is difficult to simultaneously take into account the parameter coupling characteristics and multi-source environmental effects, and it is impossible to form a complete technical system from benchmark parameter identification to cable force probability prediction during service life. It is also difficult to quantify the uncertainty of cable force prediction results.

[0005] Therefore, there is an urgent need for a Bayesian update prediction method for cable force that simultaneously considers the coupling relationship of cable-damper system parameters as well as environmental effects such as traffic, temperature, and wind, in order to improve the accuracy and reliability of cable force assessment for external dampers. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a Bayesian update prediction method for cable-damper system force considering traffic and environmental effects. This method first obtains high-precision benchmark parameters through joint identification of three parameters, and then combines multi-field monitoring data of traffic, temperature, and wind to achieve Bayesian update and probabilistic prediction of cable force during service life, effectively improving the accuracy of cable force assessment and the ability to quantify uncertainty.

[0007] The cable-damper system cable force Bayesian update prediction method provided by this invention includes the following steps:

[0008] Step 1: Establish a dynamic stiffness analysis model for the cable-damper system.

[0009] Obtain the basic parameters of the target cable-damper system, including cable length, linear mass, elastic modulus, cross-sectional area, cable inclination angle, damper installation position, damper stiffness coefficient, calculated moment of inertia of cable section, design value of damper damping coefficient, and design value of cable force.

[0010] The cable is divided into two segments according to the installation position of the damper. Based on the differential equation of the cable's transverse free vibration, the transverse dynamic stiffness element matrix of each segment considering the effects of sag, bending stiffness, tilt angle, and axial cable force is derived. The external damper is equivalent to a lumped damped stiffness element, and its contribution to the dynamic stiffness of the system is expressed as follows:

[0011] ;

[0012] In the formula, Contribution to the dynamic stiffness of the damper This is the damper stiffness coefficient. The damping coefficient of the damper. The angular frequency of vibration, It is the imaginary unit.

[0013] Based on the nodal displacement continuity condition and nodal force balance condition, the two cable segment element matrices and the damper element matrix are assembled at the damper installation node to obtain the overall dynamic stiffness matrix of the cable-damper system. The stiffness elements corresponding to the damper installation node in the matrix are superimposed with the above-mentioned damper dynamic stiffness contribution.

[0014] Setting the determinant of the overall dynamic stiffness matrix to zero, the frequency characteristic equation of the cable-damper system is established. To improve the numerical solution efficiency and stability of the complex characteristic equation, the root-finding problem of the frequency characteristic equation is transformed into an extremum optimization problem, and the following objective function is constructed:

[0015] ;

[0016] In the formula, Solve for the objective function for frequency; Let be the modal frequency to be solved, and let the angular frequency of vibration satisfy . ; For cable tension; This is a vector of known intrinsic parameters of the cable, including cable length, linear mass, elastic modulus, cross-sectional area, and cable inclination angle. Given a vector of damper parameters, including installation location and stiffness coefficient; The vector of parameters to be identified includes cable force, moment of inertia of cable section, and damper damping coefficient; To take the real part of a complex number, This indicates taking the imaginary part of a complex number; This indicates how to find the determinant of a matrix; This indicates taking the absolute value.

[0017] When the objective function reaches its maximum value, the corresponding frequency satisfies the condition of the characteristic equation. A combination of coarse frequency sweeping and fine iteration is used to solve the problem: first, frequency sweeping with a set step size initially determines the intervals in which the frequencies of each modality lie; then, within each interval, a numerical optimization algorithm searches for extreme points to obtain the theoretical modal frequency vector under given parameters.

[0018] ;

[0019] In the formula, The theoretical modal frequency vector, The m-th theoretical modal frequency, The modal order is used in the calculation.

[0020] Step 2: Joint identification of the three parameters of the cable-damper system based on the measured frequency.

[0021] Steady-state operating conditions with minimal environmental and load fluctuations, stable vibration response, and acceptable signal-to-noise ratios are selected as the benchmark operating conditions to eliminate interference from heavy traffic loads, drastic temperature changes, and strong wind disturbances on the benchmark parameter identification results. Steady-state operating conditions can be determined jointly by considering environmental parameter fluctuations, load levels, vibration response stability, and spectral signal-to-noise ratio. Specifically, a preset quantization threshold screening method or a sliding window statistical method can be used. The preset quantization threshold can be adjusted based on the cable length specifications of the target cable, the environmental characteristics of the bridge site area, and the accuracy adaptability of the monitoring system. The sliding window duration can be set according to the sampling frequency of the monitoring data and the characteristics of operating condition fluctuations.

[0022] Under baseline conditions, the acceleration response data of the target cable is acquired, and then subjected to detrending processing, filtering, fast Fourier transform, and spectral peak identification. Multi-order measured modal frequencies, verified by signal-to-noise ratio, are extracted to form a measured frequency vector.

[0023] ;

[0024] In the formula, This is the measured modal frequency vector. The m-th measured modal frequency is... The modal order is used in the calculation.

[0025] Using cable force, cable section moment of inertia, and damper damping coefficient as the parameters to be identified, a parameter vector is established:

[0026] ;

[0027] In the formula, For cable tension, Let the moment of inertia of the cable section be... is the damping coefficient of the damper.

[0028] The objective function for parameter identification is established with the goal of minimizing the weighted relative error between the measured modal frequencies and the theoretical modal frequencies:

[0029] ;

[0030] In the formula, Let be the objective function. The m-th modal frequency weight is determined based on the signal-to-noise ratio and recognition stability of each measured modal frequency. Higher weights are assigned to orders with higher signal-to-noise ratios and more stable recognition. When the recognition quality of each frequency is similar, equal weights are used by default. Let m be the theoretical modal frequency corresponding to the parameter vector to be identified. Let m be the measured modal frequency.

[0031] A global optimization algorithm is used to optimize the objective function to obtain the baseline parameter vector that minimizes the objective function. The reference cable force is obtained. Moment of inertia of reference section and reference damping coefficient .

[0032] Preferably, the identified benchmark parameters are substituted back into the dynamic stiffness analysis model, and the accuracy and physical consistency of the identification results are verified by comparing the theoretical frequency with the measured frequency.

[0033] Step 3: Bayesian update and probability prediction of traffic and environmental effects based on baseline cable force.

[0034] Based on the cable reference parameters identified in step two, a time-varying regression model of cable force is established, considering the combined effects of traffic load, temperature change, and wind load. Bayesian inference is used to jointly update the environmental sensitivity coefficient and noise variance, outputting the cable force probability prediction result. Specifically, the model includes the following sub-steps:

[0035] (1) Constructing an incremental regression model of cable force under multi-source environmental input

[0036] Acquire synchronous monitoring data at multiple sample times during the service phase of the target cable, including measured cable force, traffic load characteristics, temperature increments, and wind speed data. Using the baseline cable force identified in step two as a reference, define the cable force increment. A regression model is established between the cable force increment and multi-source environmental inputs. The cable force increment regression model can take a linear form:

[0037] ;

[0038] In the formula, Traffic load sensitivity coefficient, This is the temperature sensitivity coefficient. The wind environment sensitivity coefficient. As a measure of traffic load, For the first The temperature increment at each sample time relative to the baseline operating temperature. For wind speed, These are the residual terms; the residual terms are independent and follow the same mean of zero and variance of . The Gaussian distribution.

[0039] The above is the preferred linear regression form; when the coupling effect of multiple environmental factors needs to be considered, interaction terms such as traffic-temperature and wind speed-temperature can be introduced into the model to extend it into a coupled nonlinear regression form, and its Bayesian update solution process is consistent with the linear form.

[0040] The traffic load characterization quantity is a quantitative parameter reflecting the traffic load level on the bridge deck. It can be at least one of the following: vehicle equivalent load collected by a dynamic weighing system, vehicle traffic density monitored by video, or total vehicle weight per unit time calculated by traffic flow statistics.

[0041] (2) Define the conjugate prior distribution

[0042] The normal-inverse gamma distribution, conjugate to the Gaussian likelihood function, is selected as the environmental sensitivity coefficient vector. With noise variance The joint prior distribution; where the noise variance Obey the shape hyperparameter as Scale hyperparameters are inverse gamma distribution Environmental sensitivity coefficient vector In the given Under the condition that it follows a multivariate normal distribution:

[0043] ;

[0044] In the formula, Let be the prior mean vector of the environmental sensitivity coefficients. Let be the prior covariance scaling matrix.

[0045] Preferably, the prior mean vector Take the zero vector, the prior covariance scaling matrix By using a diagonal matrix and taking a small value for the inverse gamma prior hyperparameter, a weak prior setting is adopted overall, which can effectively avoid excessive constraints of prior information on posterior estimation.

[0046] Therefore, the joint prior of the environmental sensitivity coefficient vector and the noise variance can be expressed as:

[0047] ;

[0048] In the formula, Let be the conditional probability density function of the environmental sensitivity coefficient vector given the noise variance. It is the marginal density function of the noise variance.

[0049] (3) Update the posterior distribution based on sample data

[0050] Based on the monitoring samples, a Gaussian likelihood function is constructed. Combined with the prior distribution, the joint posterior distribution of the environmental sensitivity coefficient and noise variance is obtained analytically using Bayes' theorem.

[0051] ;

[0052] In the formula, Given environmental inputs and model parameters, the likelihood function of the observed data is used. The marginal likelihood of the observed data is the normalization constant in Bayes' theorem. Input the design matrix for the environment; This is the cable force increment response vector.

[0053] The joint posterior distribution of the environmental sensitivity coefficient and the noise variance still follows a normal-inverse gamma distribution, and the update formula for the corresponding distribution hyperparameter is as follows:

[0054] Posterior covariance matrix: ;

[0055] Posterior mean vector: ;

[0056] Posterior shape hyperparameters: ;

[0057] Posterior scale hyperparameters: ;

[0058] In the formula, Input the design matrix for the environment. This is the cable force increment response vector. This represents the number of training samples.

[0059] The posterior mean vector and posterior covariance matrix of the environmental sensitivity coefficient can be extracted from the joint posterior distribution. The former is the optimal estimate of the parameter, and the latter is used to quantify the uncertainty of the parameter estimate.

[0060] (4) Calculate the posterior prediction results of cable force

[0061] Based on the joint posterior distribution and combined with the environmental input vector at the time to be predicted, the posterior prediction distribution of the cable force increment is solved, and the mean of the cable force posterior prediction and the prediction interval at the corresponding confidence level are obtained. Finally, the cable force probability prediction result at the time to be predicted is obtained.

[0062] For any future time, the posterior predicted mean and predicted variance of the cable force increment are respectively:

[0063] ;

[0064] ;

[0065] In the formula, The input vector is used for environmental prediction at future time points. This is the updated posterior mean vector of the environmental sensitivity coefficients. The posterior covariance matrix is... , These are the shape hyperparameter and scale hyperparameter of the posterior distribution of the noise variance, respectively.

[0066] Given significance level The predicted range for the increase in cable force is:

[0067] ;

[0068] In the formula, For degrees of freedom of Distribution critical value.

[0069] The final predicted cable force at the predicted time is the sum of the baseline cable force and the predicted cable force increment:

[0070] ;

[0071] Its prediction interval is obtained by shifting the cable force increment prediction interval, thereby realizing the extension of cable force from deterministic identification to probabilistic prediction.

[0072] The present invention has the following beneficial effects:

[0073] First, a dynamic stiffness analysis model for the cable-damper system was established, taking into account the effects of cable sag, bending stiffness, inclination angle, and damper parameters. This provides a physical basis that is more consistent with the actual dynamic characteristics for predicting the cable force of cable-stayed cables with external dampers.

[0074] Second, the coupling relationship between cable force, cable section moment of inertia and damper damping coefficient is explicitly considered, and the three are used as reference parameters for joint identification. This avoids the error transfer caused by ignoring parameter coupling in traditional single-parameter cable force identification and improves the reliability of reference cable force acquisition.

[0075] Third, by adopting a weighted frequency error objective function and a global optimization strategy, the recognition accuracy and stability of different order frequencies are balanced, and the physical consistency of the parameter recognition results is strong.

[0076] Fourth, taking the benchmark cable force as a reference, the cable force changes caused by traffic load, temperature changes and wind environment are incorporated into the unified regression model. The linear superposition form is used to characterize the mapping relationship between multi-source environmental inputs and cable force increments, which can reflect the influence of multiple environmental factors on the time-varying characteristics of cable force during the bridge service stage. Those skilled in the art can introduce cross terms of environmental factors to extend it into a coupled nonlinear form according to actual engineering needs.

[0077] Fifth, the Bayesian update method is used to probabilistically model the environmental sensitivity coefficient and noise variance, and output the posterior predicted mean and posterior predicted interval of cable force, thereby quantifying the uncertainty of cable force prediction results and providing a reliable basis for subsequent state assessment and control decisions. Attached Figure Description

[0078] Figure 1 This is an overall flowchart of the method of the present invention.

[0079] Figure 2 This is a schematic diagram of the mechanical model of the cable-damper system.

[0080] Figure 3 This is the posterior marginal distribution plot of the environmental sensitivity coefficient.

[0081] Figure 4 This is the posterior marginal distribution of the noise variance.

[0082] Figure 5 A time history diagram for posterior prediction of cable force considering traffic and environmental effects. Detailed Implementation

[0083] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the technical solutions and implementation processes of the present invention, and are not intended to limit the scope of protection of the present invention.

[0084] Example

[0085] To verify the effectiveness and feasibility of the method of this invention, a long cable with an external damper of a long-span cable-stayed bridge was used as a simulation prototype, and a complete verification was carried out through numerical simulation. Figure 2 As shown, one end of the target cable is anchored to the bridge tower, and the other end is anchored to the main beam. An external viscous damper is installed near the anchorage position at the beam end. The basic parameters of the target cable and damper are as follows: cable length... =104.5 m, cable mass is 72.6 kg / m, cable inclination angle =30.3°, elastic modulus is 20×105 MPa, cross-sectional area 0.110 m² 2 The moment of inertia of the cross section is calculated as follows: =5.71×10 -6 m 4 The design cable force is =2389 kN; the damper is installed 2.93 m from the beam end, and the damper stiffness coefficient is... =50 kN / m, damping coefficient design value =267 kN·s / m.

[0086] like Figure 1 As shown in the example, a multi-field coupled cable force Bayesian prediction method for a cable-damper system is provided, comprising the following steps:

[0087] Step 1: Establish a dynamic stiffness analysis model for the cable-damper system.

[0088] Based on the differential equation of lateral free vibration of the cable, the cable is divided into an upper cable segment (from the damper to the tower end) and a lower cable segment (from the beam end to the damper) according to the installation position of the damper. The lateral dynamic stiffness element matrix for each cable segment, considering the effects of sag, bending stiffness, tilt angle, and axial cable force, is derived. The external damper is equivalent to a lumped damped stiffness element, and its contribution to the dynamic stiffness of the system is expressed as:

[0089] ;

[0090] In the formula, Contribution to the dynamic stiffness of the damper This is the damper stiffness coefficient. The damping coefficient of the damper. The angular frequency of vibration, It is the imaginary unit.

[0091] Based on the nodal displacement continuity condition and the nodal force equilibrium condition, the dynamic stiffness element matrix of the two cable segments and the lumped element matrix of the damper are assembled at the damper installation node to obtain the overall dynamic stiffness matrix of the cable-damper system. During assembly, the stiffness elements of the corresponding damper mounting nodes in the matrix are superimposed with the aforementioned contribution of the damper's dynamic stiffness.

[0092] Setting the determinant of the overall dynamic stiffness matrix to zero, we establish the frequency characteristic equation in complex form. To improve the efficiency and stability of the numerical solution of the complex characteristic equation, the root-finding problem is transformed into an extremum optimization problem, and the objective function for frequency solution is constructed as follows:

[0093] ;

[0094] In the formula, Solve for the objective function for frequency; Let be the modal frequency to be solved, and let the angular frequency of vibration satisfy . ; For cable tension; This is a vector of known intrinsic parameters of the cable, including cable length, linear mass, elastic modulus, cross-sectional area, and cable inclination angle. Given a vector of damper parameters, including installation location and stiffness coefficient; The vector of parameters to be identified includes cable force, moment of inertia of cable section, and damper damping coefficient; To take the real part of a complex number, This indicates taking the imaginary part of a complex number; This indicates how to find the determinant of a matrix; This indicates taking the absolute value.

[0095] A two-step method of coarse frequency sweep followed by fine iteration is employed: first, a frequency sweep with a step size of 0.01 Hz is used to initially determine the frequency range of each modality; then, Newton's iteration method is used within each range to find the extreme points and calculate the theoretical modal frequency vector under given parameters. ,in The modal frequency order used in the calculation is 8 in this embodiment, which is consistent with the measured frequency order.

[0096] Step Two Implementation: Joint Identification of Three Parameters of the Cable-Damper System

[0097] Based on bridge service monitoring data, the system continuously outputs vibration acceleration time history, traffic load characteristics, ambient temperature, and wind speed data for the target cable. A baseline period is selected according to baseline operating conditions: the wind speed must be below 3 m / s, the temperature change rate below 2 ℃ / h, and the traffic load characteristics below 30% of the daily average for a continuous period of at least 2 hours. The daily average is the arithmetic mean of the traffic load characteristics of the bridge deck where the target cable is located over the past 7 consecutive natural days. The extracted modal frequency signal-to-noise ratio is greater than 10 dB.

[0098] The above threshold values ​​are preferred values ​​for the long cables of the long-span cable-stayed bridge in this embodiment. Those skilled in the art can adaptively adjust the threshold values ​​for judging steady-state conditions based on the actual bridge type, cable length, monitoring system accuracy, and regional environmental characteristics. This embodiment selects the steady-state early morning period as the baseline condition, during which traffic load is close to zero, ambient temperature is stable, and wind speed disturbance is small, which meets the requirements of the baseline condition.

[0099] Gaussian white noise with an amplitude of 5% of the peak acceleration response was added to the cable acceleration time history during the reference period to simulate the engineering measurement environment. After detrending processing, bandpass filtering, fast Fourier transform, and spectral peak identification, the 8th effective measured modal frequencies were extracted to form the measured frequency vector. .

[0100] With cable force Moment of inertia of cable section and damper damping coefficient As the parameters to be identified, establish a vector of parameters to be identified:

[0101] ;

[0102] The objective function for parameter identification is established with the goal of minimizing the weighted relative error between the measured modal frequencies and the theoretical modal frequencies:

[0103] ;

[0104] In the formula, For the first The frequency weights of each order are equal. In this embodiment, the signal-to-noise ratios of the effective modes of each order meet the requirements and the recognition stability is similar. Therefore, equal weights are used and the weights of each order are equal.

[0105] A genetic algorithm is used to globally optimize the objective function. The algorithm control parameters are set as follows: population size 300, crossover probability 0.75, mutation probability 0.05, and an elitist retention strategy. The baseline parameter vector is obtained after iterative convergence. Among them, the reference cable force =2365 kN, moment of inertia of the reference section is =5.81×10 -6 m 4 Reference damping coefficient =270.3 kN·s / m.

[0106] Frequency verification implementation

[0107] The identified baseline parameter vector Substituting the dynamic stiffness analysis model established in step one back into the calculation, the theoretical modal frequencies of each order under the reference working condition were calculated and compared with the measured modal frequencies. The results are shown in Table 1. As can be seen from the table, the maximum relative error of each frequency order is 1.41%, which meets the engineering accuracy requirements, indicating that the joint identification results of the three parameters have good physical consistency.

[0108] Table 1 Comparison of theoretical and measured modal frequencies under reference operating conditions

[0109] 1 0.919 0.906 1.41 2 1.801 1.792 0.50 3 2.710 2.697 0.48 4 3.616 3.604 0.33 5 4.535 4.516 0.42 6 5.460 5.433 0.49 7 6.390 6.358 0.50 8 7.333 7.291 0.57

[0110] Step 3 Implementation: Bayesian Update and Probabilistic Prediction of Cable Force Based on Benchmark Cable Force

[0111] The reference cable force obtained in step two As a reference benchmark, environmental effect modeling and Bayesian probability prediction of cable stress during service life were carried out. The specific process is as follows:

[0112] (1) Constructing an incremental regression model of cable force under multi-source environmental input

[0113] Six consecutive hours of synchronous simulation monitoring data were selected, and a sample set was constructed using a uniform 5-minute time interval. The first 4 hours and 48 minutes of data were used as training samples, and the remaining 1 hour and 12 minutes of data were used as test samples. The cable force increment was defined. Establish a regression model between cable force increment and multi-source environmental input:

[0114] ;

[0115] In the formula, Traffic load sensitivity coefficient, This is the temperature sensitivity coefficient. The wind environment sensitivity coefficient. As a measure of traffic load, For the first The temperature increment at each sample time relative to the baseline operating temperature. For wind speed, These are the residual terms; the residual terms are independent and follow the same mean of zero and variance of . The Gaussian distribution.

[0116] (2) Define the conjugate prior distribution

[0117] The normal-inverse gamma distribution, conjugate to the Gaussian likelihood function, is selected as the environmental sensitivity coefficient vector. With noise variance Joint prior distribution:

[0118] ;

[0119] noise variance It follows an inverse gamma prior distribution: Among them, shape hyperparameters Scale hyperparameters Weak prior settings are used;

[0120] Given the noise variance, the environmental sensitivity coefficient vector follows a multivariate normal distribution: , where the prior mean vector Take the zero vector, the prior covariance scaling matrix Take the diagonal matrix .

[0121] (3) Update the posterior distribution based on the training samples

[0122] Based on the training samples, a Gaussian likelihood function is constructed. Combining the aforementioned conjugate prior, the joint posterior distribution of the environmental sensitivity coefficient and noise variance is obtained analytically based on Bayes' theorem. Its posterior form is still a normal-inverse gamma distribution. The parameter update formula is as follows:

[0123] Posterior covariance matrix: ;

[0124] Posterior mean vector: ;

[0125] Posterior shape hyperparameters: ;

[0126] Posterior scale hyperparameters: ;

[0127] In the formula, Input the design matrix for the environment. This is the cable force increment response vector. This represents the number of training samples.

[0128] The posterior mean vector and posterior covariance matrix of the environmental sensitivity coefficients are extracted from the joint posterior distribution. The former represents the optimal estimate of the parameters, while the latter is used to quantify the uncertainty of the parameter estimates. The posterior distribution of the parameters is as follows: Figure 3 , Figure 4 As shown. The updated results indicate that traffic load contributes positively to the increase in cable force, while temperature rise contributes negatively, which is consistent with the physical laws governing cable stress. The posterior distribution of the wind load sensitivity coefficient is concentrated near zero, indicating that the contribution of wind speed effect to cable force change is relatively insignificant under the current operating conditions.

[0129] (4) Calculate the posterior prediction results of cable force

[0130] Based on the joint posterior distribution, the predicted input vector is combined with the corresponding environment for the time to be predicted within the test period. Solve for the posterior predicted distribution of the cable force increment, and its predicted mean and predicted variance are as follows:

[0131] ;

[0132] ;

[0133] Given significance level At a 95% confidence level, the predicted range for the increase in slack is:

[0134] ;

[0135] In the formula, For degrees of freedom of Distribution critical value.

[0136] Ultimately, the predicted cable force at the predicted time is the sum of the baseline cable force and the predicted cable force increment, i.e. Its prediction interval is obtained by shifting the cable force increment prediction interval. The prediction result time history of the test sample is as follows: Figure 5 As shown. By Figure 5 It can be seen that all simulated and measured cable force points fall within 95% of the prediction range during the test period, with a range coverage of 100%, indicating that the method has good prediction accuracy and uncertainty envelope capability.

Claims

1. A multi-field coupled Bayesian prediction method for cable-damper systems, characterized in that, Includes the following steps: Step 1: Establish a dynamic stiffness analysis model for the cable-damper system; obtain the basic parameters of the target cable-damper system, and construct an overall dynamic stiffness matrix considering sag effect, bending stiffness, cable inclination angle and damper parameters; establish and solve the frequency characteristic equation based on the overall dynamic stiffness matrix to obtain the theoretical modal frequency vector. Step 2: Joint identification of three parameters of the cable-damper system based on measured frequencies; select steady-state condition as the reference condition, and obtain the multi-order measured modal frequencies of the target cable under the reference condition; take cable force, cable section moment of inertia and damper damping coefficient as the parameters to be identified, and optimize with the goal of minimizing the weighted relative error between the measured modal frequency and the theoretical modal frequency to obtain the reference parameter vector; Step 3: Bayesian update and probabilistic prediction of traffic and environmental effects based on baseline cable force; Using the baseline cable force as a reference, the cable force increment is defined, and a multi-source environmental input regression model is established between the cable force increment and traffic load, temperature increment, and wind speed. The parameters of the regression model are updated using Bayesian methods to solve the posterior prediction distribution of the cable force increment, thereby obtaining the predicted mean and prediction interval of the cable force. The final predicted value of the cable force is the sum of the baseline cable force and the predicted value of the cable force increment.

2. The method according to claim 1, characterized in that, In step one, the determinant of the overall dynamic stiffness matrix is ​​set to zero to establish the frequency characteristic equation, transforming the root-finding problem into an extremum optimization problem and constructing the frequency solution objective function. The objective function is: ； In the formula, Solve for the objective function for frequency; Let be the modal frequency to be solved, and let the angular frequency of vibration satisfy . ; For cable tension; This is a vector of known intrinsic parameters of the cable, including cable length, linear mass, elastic modulus, cross-sectional area, and cable inclination angle. Given a vector of damper parameters, including installation location and stiffness coefficient; The vector of parameters to be identified includes cable force, moment of inertia of cable section, and damper damping coefficient; To take the real part of a complex number, This indicates taking the imaginary part of a complex number; This indicates how to find the determinant of a matrix; This indicates taking the absolute value.

3. The method according to claim 1, characterized in that, The steady-state conditions described in step two are filtered through preset quantitative conditions, including a monitoring period duration threshold, a wind speed threshold, an ambient temperature change rate threshold, a traffic load relative level threshold, and a modal frequency signal-to-noise ratio threshold. The daily average value for the same period is the arithmetic mean of the traffic load characteristics of the corresponding time period for a preset number of consecutive natural days on the bridge deck where the target cable is located.

4. The method according to claim 1, characterized in that, The steady-state condition described in step two is determined by the sliding window statistical method: the monitoring data of the entire time period are traversed by a sliding window of a set duration, the coefficient of variation of environmental parameters, the coefficient of variation of load parameters and the signal-to-noise ratio of vibration response spectrum are calculated in each window, and the continuous time period with the smallest comprehensive fluctuation and the signal-to-noise ratio meeting the preset requirements is selected as the benchmark condition.

5. The method according to claim 1, characterized in that, The objective function for parameter identification used in step two is: ； In the formula, Let be the objective function. The parameter vector to be identified. For the first First-order modal frequency weights, The first parameter corresponding to the parameter to be identified First-order theoretical modal frequencies, For the first Measured modal frequencies of the first order. The modal order is used in the calculation.

6. The method according to claim 1, characterized in that, The cable force incremental regression model described in step three adopts a linear form, and its expression is: ； In the formula, Traffic load sensitivity coefficient, This is the temperature sensitivity coefficient. The wind environment sensitivity coefficient. As a measure of traffic load, For the first The temperature increment at each sample time relative to the baseline operating temperature. For wind speed, These are the residual terms; the residual terms are independent and follow the same mean of zero and variance of . The Gaussian distribution.

7. The method according to claim 1, characterized in that, Step three employs a normal-inverse gamma conjugate prior distribution for Bayesian updating; where the noise variance... Follows inverse gamma prior distribution Given the noise variance, the environmental sensitivity coefficient vector follows a multivariate normal distribution. Prior mean vector Take the zero vector, the prior covariance scaling matrix By using a diagonal matrix and a weak prior setting, we can avoid excessive constraints on posterior estimations imposed by prior information.

8. The method according to claim 7, characterized in that, The parameter update formula for the joint posterior distribution mentioned in step three is: Posterior covariance matrix: ; Posterior mean vector: ; Posterior shape hyperparameters: ; Posterior scale hyperparameters: ; In the formula, Input the design matrix for the environment. This is the cable force increment response vector. This represents the number of training samples.

9. The method according to claim 7, characterized in that, In step three, the posterior predicted mean and predicted variance of the cable force increment are respectively: ； ； In the formula, The input vector is used for environmental prediction at future time points; The predicted range for the cable force increment is: ； In the formula, For degrees of freedom The critical value of the Student-t distribution, The significance level is [value]; the final cable force prediction value is [value]. .

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