Nonlinear aerodynamic damping self-supervised identification method for long-span bridges based on hardened non-gaussian response probability density constraint

Abstract

The application relates to a long-span bridge nonlinear aerodynamic damping self-supervised identification method based on a hardened non-Gaussian response probability density constraint, which comprises the following steps: S1, acquiring displacement response data of a structure under side wind excitation; S2, pre-processing the displacement response data and constructing a target displacement probability density function; S3, establishing a motion equation related to first-order modal response containing nonlinear aerodynamic damping; S4, deducing an analytical displacement probability density function based on a Fokker-Planck-Kolmogorov equation; S5, constructing a probability density function information embedded self-supervised neural network PDF-ThetaNet; S6, constructing a composite probability distribution loss function and training the PDF-ThetaNet; and S7, outputting nonlinear aerodynamic damping parameters and buffeting excitation strength of a long-span bridge structure. Through construction of a nonlinear aerodynamic damping and unknown buffeting force spectrum strength identification model and based on vertical displacement, a self-supervised neural network based on a probability density function constraint is used to realize identification of nonlinear aerodynamic damping and buffeting force power spectrum strength.

Description

Technical Field

[0001] This invention belongs to the field of bridge wind engineering and structural health monitoring technology, and relates to a self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints. In particular, it relates to a method for identifying amplitude-dependent nonlinear aerodynamic damping of long-span bridge structures based on a probability density function-constrained self-supervised neural network. Background Technology

[0002] Long-span bridges typically exhibit low natural frequencies, low structural damping, and high overall flexibility, making them prone to significant wind-induced vibrations under wind loads. Among these wind-induced vibrations, vortex-induced vibrations require particular attention. This type of vibration is closely related to aerodynamic damping effects and can induce large-amplitude oscillations near vortex-locked wind speeds, thereby affecting the structure's normal service performance, fatigue performance, and operational safety. Aerodynamic damping near vortex-locked wind speeds is not a fixed structural property but rather an aerodynamic parameter that varies with the vibration amplitude; it may exhibit negative damping characteristics and show significant amplitude-dependent nonlinearity. This nonlinear aerodynamic damping alters the statistical characteristics of the structural response, causing it to exhibit a hardened non-Gaussian distribution that is difficult to characterize completely by a single resonant process or an ideal Gaussian random process. Therefore, accurately identifying amplitude-dependent nonlinear aerodynamic damping is crucial for assessing the vortex-induced vibration performance of flexible wind-sensitive structures, predicting wind-induced responses, and designing for wind-resistant safety.

[0003] However, existing technologies mostly focus on nonlinear aerodynamic damping identification based on random response time histories or response probability density functions, and have not yet effectively met the requirements of label-free, joint inversion of amplitude-dependent nonlinear aerodynamic damping parameters and unknown buffeting force power spectrum intensity for long-span bridge structures under random wind loads during vortex-induced vibration. While existing random vibration identification methods can extract aerodynamic damping information from response data to some extent, the random decrement method is mainly applicable to linear damping systems and struggles to accurately handle nonlinear damping cases. Methods based on response statistical moments rely on standard deviation, kurtosis, and moment transformation relationships, limiting their accuracy in characterizing the relationship between higher-order statistics and nonlinear damping parameters. Nonlinear identification methods based on Kalman filtering are highly sensitive to process noise and measurement noise matrix settings; inaccurate noise assumptions can easily lead to increased estimation errors or even filter divergence. On the other hand, identification methods based on response probability density functions typically require simultaneous estimation of multiple potential parameters, such as nonlinear aerodynamic damping coefficients and fluctuating wind load spectrum intensity, which can easily lead to parameter non-uniqueness issues. When the order of the aerodynamic damping model increases or the available structural damping conditions are insufficient, different parameter combinations may yield similar response probability density functions, but the corresponding aerodynamic damping results are inconsistent, thus limiting the reliability and engineering applicability of such methods. Therefore, there is an urgent need for a label-free inversion method for nonlinear aerodynamic damping oriented towards hardened non-Gaussian random responses. This method would embed the statistical physical constraints inherent in the response probability density function into the neural network training process, construct a joint identification model of nonlinear aerodynamic damping parameters and unknown buffeting force spectrum intensity, and achieve the collaborative inversion of amplitude-dependent nonlinear aerodynamic damping and unknown buffeting force spectrum intensity based on the measured displacement response probability density function, thereby improving the stability, uniqueness, and practicality of nonlinear aerodynamic damping identification under actual wind conditions. Summary of the Invention

[0004] In view of this, in order to solve the problems of existing amplitude-dependent nonlinear aerodynamic damping identification methods under vortex-induced vibration of long-span bridges, this invention addresses the shortcomings of existing methods. These methods suffer from drawbacks such as the difficulty of adapting the random subtraction method to nonlinear damping, the limited accuracy of high-order statistical quantities characterized by the statistical moment method, and the sensitivity of the Kalman filter method to noise assumptions. Furthermore, they face the problems of parameter non-uniqueness, insufficient reliability, and insufficient engineering applicability when identifying based on response probability density. This makes it impossible to collaboratively invert nonlinear aerodynamic damping parameters and unknown buffeting force power spectrum intensity under label-less conditions, and it is difficult to meet the stability, uniqueness, and practicality requirements of identification under actual wind conditions. Based on the above-mentioned problems of existing technologies, this invention provides a self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints. This identification method has simple steps and a reasonable design. It constructs an identification model for nonlinear aerodynamic damping and unknown buffeting force spectrum intensity, and uses a self-supervised neural network based on probability density function constraints to identify nonlinear aerodynamic damping and buffeting force power spectrum intensity based on vertical displacement.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints includes the following steps:

[0007] S1. Obtain the vertical displacement response data of the structural components under crosswind excitation in the crosswind direction, which is used to represent the generalized displacement response corresponding to the first-order mode of the structure.

[0008] S2. Preprocess the vertical displacement response data from step S1 to obtain a stationary random displacement response sequence for parameter identification; divide the stationary random displacement response sequence into several data segments and obtain the target displacement probability density function using a density estimation method; simultaneously, use multiple structural damping conditions for joint identification, and construct the target displacement probability density function for the i-th condition respectively. ;

[0009] S3. Establish the equations of motion related to the first-order modal response, which include nonlinear aerodynamic damping. The equations of motion related to the first-order modal response are expressed as follows:

[0010]

[0011] in, Indicates generalized structural mass; Indicates generalized displacement; Indicates the inherent structural damping ratio; Represents the natural circular frequency corresponding to the fundamental mode of the structure; This represents the first generalized force, which is composed of the flutter force. and self-excitation The sum of them constitutes, that is ;

[0012] S4. Substitute the van der Pol transform into the motion equations related to the first-order modal response in step S3 to obtain the stochastic differential equations controlling the amplitude and phase processes. Use the Wong–Zakai correction to transform the stochastic differential equations into Itô form. Perform periodic averaging over one vibration period to obtain the average Itô stochastic differential equation for the amplitude process. Based on this, establish the one-dimensional Fokker–Planck–Kolmogorov equation. Under zero-probability boundary conditions, solve the Fokker–Planck–Kolmogorov equation to obtain the amplitude probability density function. Based on the amplitude probability density function... The analytical displacement probability density function of the displacement response is obtained through integral transformation;

[0013] S5. Output parameters of the constructed PDF-ThetaNet Substituting the analytical displacement probability density function obtained in step S4, we obtain the predicted displacement probability density function. and the target displacement probability density function obtained in step S2 By comparison, self-supervised identification under the condition of no aerodynamic damping parameter labeling is achieved;

[0014] S6. To simultaneously constrain the principal probability density region and harden the non-Gaussian tail region, a composite probability distribution loss function is constructed, and the PDF-ThetaNet constructed in step S5 is trained.

[0015] S7. Substitute the parameter vector obtained after training in step S6 into the nonlinear aerodynamic damping model to output the amplitude-dependent nonlinear aerodynamic damping parameters and flutter excitation intensity of the long-span bridge structure.

[0016] The beneficial effects of this invention are as follows:

[0017] 1. Existing neural network-based identification methods typically rely on a large amount of known parameters or labeled data for supervised training. However, in actual bridge wind-induced vibration environments, nonlinear aerodynamic damping parameters are often deeply embedded in complex fluid-structure interaction effects, making them difficult to measure directly through experiments or sensors. To address this challenge, this invention innovatively utilizes the probability density function estimated from the displacement response itself to construct a self-supervised constraint mechanism. This means that the network no longer needs real damping parameters as labels during training; instead, it optimizes directly by comparing the predicted and actual response distributions. This is perfectly suited for "labelless" scenarios in practical engineering where parameters cannot be directly observed, significantly reducing the difficulty and cost of data acquisition.

[0018] 2. To address the problem of unstable identification or even failure of existing identification methods under unknown buffeting force spectrum conditions, this invention proposes a synchronous inversion solution. This invention cleverly incorporates the buffeting force power spectrum intensity as a potential physical quantity to be identified, integrating it into the output parameters of a neural network, and performing synchronous joint inversion with amplitude-dependent nonlinear aerodynamic damping parameters. This design breaks the limitation of traditional methods that require pre-given or assumed complete wind load spectra, enabling this invention to maintain high robustness and identification accuracy even in complex and variable wind load environments in practical engineering projects.

[0019] 3. Under vortex-induced vibration, the response of long-span bridges typically exhibits a "hardened non-Gaussian" characteristic, falling between a pure harmonic process and a Gaussian random process. Traditional linear methods often struggle to fully utilize this unique statistical information. This invention, based on the Fokker-Planck-Kolmogorov (FPK) equations, rigorously derives and establishes an analytical displacement probability density function. This model successfully links amplitude-dependent nonlinear aerodynamic damping parameters directly to the probability distribution of the structural response, enabling the network to capture damping variation patterns from subtle non-Gaussian distribution features. This significantly improves the ability to identify nonlinear aerodynamic damping and enhances its physical interpretability.

[0020] 4. Conventional mean squared error (MSE) loss based on probability density functions is often dominated by the high-density main probability peak region, making it difficult for the probability density tail information representing large-amplitude events to effectively participate in optimization. To address this, this invention designs a composite loss function that integrates log-PDF loss, original probability density loss (PDFLoss), and cumulative distribution function loss (CDF Loss). This multi-layered constraint mechanism enables the network to accurately fit the main peak region of the probability distribution while enhancing its sensitivity to long-tail features in hardened non-Gaussian responses, thereby comprehensively improving the accuracy of parameter identification and the model's generalization ability.

[0021] 5. This invention offers exceptional convenience in data acquisition, utilizing only the vertical displacement response data of the structure to complete the identification task. It does not rely on difficult-to-measure direct aerodynamic data, nor does it require simultaneous acquisition of velocity, acceleration, or complete wind load time histories, greatly simplifying the hardware configuration and data processing flow of the testing system. This feature makes this invention suitable not only for wind tunnel tests under strictly controlled conditions but also for seamless integration with actual bridge structural health monitoring (SHM) systems equipped only with conventional displacement gauges, demonstrating broad prospects for engineering application.

[0022] 6. Unlike traditional methods that require significant computational resources for large-scale fluid-structure interaction numerical simulations or complex stochastic search optimizations, this invention combines the analytical FPK probability density function with a lightweight neural network architecture (PDF-ThetaNet). This hybrid physics-driven and data-driven strategy significantly reduces computation time while maintaining high accuracy. It can quickly identify nonlinear aerodynamic damping parameters and buffeting force power spectrum intensity, meeting the rapid feedback needs of engineering designers in wind-induced vibration assessment and possessing excellent engineering practical value.

[0023] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0024] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0025] Figure 1 This is a flowchart illustrating the PDF-ThetaNet architecture and training process of this invention.

[0026] Figure 2 This is a flowchart of the self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints according to the present invention.

[0027] Figure 3 This is a cross-sectional view of a bridge segment model used in a wind tunnel test according to an embodiment of the present invention;

[0028] Figure 4 This is a comparison of the displacement PDF, aerodynamic damping, and chattering excitation intensity identified in the embodiments of the present invention with the theoretical values, wherein... Figure 4 (a) For converted wind speed Comparison chart of identified and target values ​​of the bridge displacement PDF; Figure 4 (b) For converted wind speed Comparison chart of identified and target values ​​of the bridge displacement PDF; Figure 4 (c) is the converted wind speed Comparison chart of identified and target values ​​of the bridge displacement PDF; Figure 4 (d) is the converted wind speed Comparison chart of identified and target values ​​of the bridge displacement PDF; Figure 4 (e) represents the buffeting excitation intensity under different equivalent wind speeds. A comparison chart of the identified value and the target value; Figure 4 (f) represents the nonlinear aerodynamic damping ratio under different equivalent wind speeds. A comparison chart of the identified value and the target value. Detailed Implementation

[0029] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0030] like Figure 1 The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints, as shown, includes the following steps:

[0031] S1. Obtain displacement response data of structural components under crosswind excitation, wherein the structural components are main beams, bridge towers, or other wind-sensitive flexible components of long-span bridges subjected to crosswinds; the displacement response data can be obtained by wind tunnel testing, on-site structural health monitoring systems, or numerical simulation.

[0032] In practice, the vertical displacement response data of the structural components under crosswind excitation is obtained and denoted as follows: , representing the generalized displacement response corresponding to the first mode of the structure.

[0033] To eliminate the influence of structural dimensions on subsequent aerodynamic damping identification, the vertical displacement response data were dimensionless:

[0034]

[0035] in, Represented as dimensionless displacement, This indicates the characteristic width of the main beam of the bridge or the structure to be identified.

[0036] S2. Preprocess the displacement response data and construct the target displacement probability density function; that is, perform mean removal and outlier elimination on the displacement response data obtained in step S1 to obtain a stationary random displacement response sequence for parameter identification. Divide the stationary random displacement response sequence into several data segments, and obtain the target displacement probability density function through kernel density estimation, histogram normalization, or other probability density estimation methods. , This represents the probability density function of the target displacement estimated from measured or experimental displacement response data.

[0037] Multiple structural damping conditions are used for joint identification, and the target displacement probability density function is constructed for the i-th condition respectively:

[0038]

[0039] in, This indicates the number of response conditions participating in the joint identification.

[0040] S3. Establish the motion equations related to the first-order modal response, which includes nonlinear aerodynamic damping;

[0041] For a flexible structure subjected to crosswind excitation, the equation of motion related to the first-order modal response can be expressed as:

[0042]

[0043] in, Indicates generalized structural mass; Indicates generalized displacement; Indicates the inherent structural damping ratio; Represents the natural circular frequency corresponding to the fundamental mode of the structure; This represents the first generalized force, which is composed of the flutter force. and self-excitation The sum of them constitutes, that is .

[0044] Self-excitation force Nonlinear aerodynamic damping Control, represented as:

[0045]

[0046] Nonlinear aerodynamic damping Represented as a polynomial function of dimensionless displacement:

[0047]

[0048] in, air density; The width of the structure; The effective mass per unit height of the structure; The coefficients of the aerodynamic damping model to be identified;

[0049] Flutter force In practical engineering applications, this is usually unknown; therefore, the equivalent stochastic excitation under the unit generalized mass is defined as:

[0050]

[0051] And Modeled as a Gaussian white noise process in the Stratoovich sense, its one-sided power spectral density at the fundamental frequency of the structure is denoted as... .

[0052] S4. Derive the analytical displacement probability density function based on the Fokker–Planck–Kolmogorov equation;

[0053] In this embodiment, the structural response near the vortex-induced locking region can be considered as a narrowband random process, therefore a van der Pol transform is introduced:

[0054]

[0055] in, For dimensionless displacement, The characteristic width of the structure; and It is a generalized harmonic function; , ,and It is a stochastic process, where Represents the phase process, Indicates the amplitude process;

[0056] By substituting the van der Pol transformation, i.e., equations (4a)–(4c), into the original equation of motion, i.e., equation (1), and differentiating with respect to time t, the stochastic differential equations controlling the amplitude and phase processes are obtained as follows:

[0057]

[0058] in, It is modeled as Gaussian white noise in the Stratoovich sense; and Here are the drift term coefficients, representing the effects of structural damping and nonlinear aerodynamic damping on amplitude and phase, respectively; and The diffusion coefficient is used to characterize the random disturbance caused by the buffeting force.

[0059]

[0060] In stochastic vibration analysis, the Fokker–Planck–Kolmogorov (FPK) equations are used to describe the time evolution of the probability density function (PDF) of the system's state variables; because Modeled as Gaussian white noise in the Stratoovich sense, the stochastic differential equations in equations (5a) and (5b) should first be transformed into their Itô form using the Wong–Zakai correction:

[0061]

[0062] in, Represents the standard Wiener process; corrected drift coefficients and diffusion covariance term for:

[0063]

[0064] in, Taken as calculated at the structure's natural frequency. The one-sided power spectral density.

[0065] Based on equations (7a) and (7b), a periodic average is performed on a vibration cycle, i.e. The average Itô stochastic differential equation for the amplitude and phase process can be obtained as follows:

[0066]

[0067] in:

[0068]

[0069] The FPK equation corresponding to the average Itô stochastic differential equation in equations (9a) and (9b) can be written as follows:

[0070]

[0071] in, This represents the joint probability density function of the amplitude-phase state variables.

[0072] amplitude and phase They are approximately independent. Therefore, the amplitude process It can be viewed as a one-dimensional Markov diffusion process, with its probability density... satisfy

[0073]

[0074] Under stationary conditions, the probability density does not change with time, i.e. .exist hour and Under the boundary condition approaching zero, regarding both sides of equation (12) with respect to... Integral, to obtain

[0075]

[0076] By introducing Equation (13) can be transformed into an ordinary differential equation. The solution is

[0077]

[0078] in, This is the normalization constant.

[0079] Displacement response The probability density function can be derived from the amplitude probability density function. get:

[0080]

[0081] S5. Construct a self-supervised neural network PDF-ThetaNet with embedded probability density function information;

[0082] Based on the analytical displacement PDF formula (Equation 15) derived in step S4, a PDF information neural network, called PDF-ThetaNet, is developed to identify nonlinear aerodynamic damping parameters. A second-order aerodynamic damping model (i.e., Equation (3)) is adopted, and the unknown parameter vector is defined as... Specifically, PDF-ThetaNet takes the initial guesses of this parameter vector as input. Then, the generated parameters are substituted into the analytical displacement PDF model. In the above, i.e., Equation (15). By minimizing the difference between the analytical displacement PDF and the target displacement PDF, the proposed framework embeds the physical relationship between nonlinear aerodynamic damping and non-Gaussian response statistical characteristics into the training process, thereby achieving label-free aerodynamic damping identification through PDF-level consistency rather than parameter-level supervision.

[0083] The PDF-ThetaNet architecture consists of initial parameter inputs, an input projection layer, a stacked residual multilayer perceptron (MLP) module, and an output projection layer. Initial guessed parameter vectors. The data is first projected into a high-dimensional space and then processed by a residual MLP module. This residual MLP module consists of layer normalization, SiLU activation functions, and fully connected layers. Each module employs a residual update approach, which stabilizes the feature transformation and improves convergence performance. The output projection layer generates the estimated parameter vector. Subsequently, these parameters are substituted into the analytical displacement PDF model for loss evaluation. The trainable parameters are optimized using a two-stage strategy: first, robust initial optimization using Adam (500 epochs), followed by local fine-tuning using L-BFGS (100 epochs). The detailed network architecture and training process of PDF-ThetaNet are shown in Figure 2.

[0084] S6. Construct a composite probability distribution loss function and train PDF-ThetaNet;

[0085] The loss function is constructed by forcing consistency between the displacement PDF predicted by PDF-ThetaNet and the target displacement PDF obtained from the response data. Since the probability density values ​​of the main distribution region and the tail region can differ by several orders of magnitude, the traditional PDF-based mean square error (MSE) is often dominated by the high-density main peak region, resulting in a small contribution of low-probability tail errors to the optimization process. However, nonlinear aerodynamic damping parameters primarily affect the non-Gaussian tail behavior of the response distribution. Therefore, a logarithmic log-PDF loss is introduced to enhance sensitivity to relative errors in the tail region.

[0086] However, using only log-PDF loss may overemphasize the tail region, which is often affected by statistical fluctuations due to finite samples. To maintain the fitting accuracy of the main probability density region, the original PDF loss is also introduced. Furthermore, a cumulative distribution function (CDF) loss is introduced to constrain the cumulative probability, thus ensuring the consistency of the overall probability distribution, not just the consistency of local PDF values. Therefore, the total loss function is defined as:

[0087]

[0088] in, , and Let represent the log-PDF loss, the original PDF loss, and the CDF loss, respectively. The corresponding weighting coefficients are set to... , , and .

[0089] Log probability density loss, or log-PDF loss, is expressed as:

[0090]

[0091] in, The number of fitting operating conditions; To determine the number of grid points for the displacement, this method selects to divide the displacement into 2048 grid points; For the first Under the first working condition Normalized displacement-related weights at each displacement grid point; and They represent the first The predicted displacement PDF and target displacement PDF for each working condition; Used to avoid the logarithm from being zero.

[0092] To further enhance the contribution of the large amplitude response region, displacement-related weights are introduced:

[0093]

[0094] in, For minimum weight, Control the degree of emphasis on the tail. This weight is further normalized to...

[0095]

[0096] The original probability density loss (original PDF loss) and the cumulative distribution function loss (CDF loss) are defined as follows:

[0097]

[0098] CDF is obtained by integrating the corresponding PDF:

[0099]

[0100] The Adam optimizer is used for initial optimization of PDF-ThetaNet, followed by local fine-tuning using the L-BFGS optimizer to obtain the final converged parameters to be identified.

[0101] S7 outputs the nonlinear aerodynamic damping parameters and flutter excitation intensity of the long-span bridge structure.

[0102] The result obtained after step S6 training is completed. Substituting into the nonlinear aerodynamic damping model, we obtain the amplitude-dependent nonlinear aerodynamic damping curve of the structure:

[0103] (twenty three)

[0104] Simultaneously output equivalent chattering excitation intensity

[0105] In this embodiment, the It can be used to evaluate the vortex-induced vibration response, amplitude-dependent aerodynamic damping characteristics, and wind-induced vibration safety performance of long-span bridge structures under crosswind excitation; This method can be used to characterize the equivalent intensity of unknown random buffeting excitation at the fundamental frequency of a structure. The method of this invention does not require prior knowledge of the actual aerodynamic damping parameter labels or a complete buffeting force spectrum; it only requires structural displacement response data to jointly identify the nonlinear aerodynamic damping parameters and the equivalent buffeting excitation intensity. When it is necessary to improve the stability of the identification, response data from two or more different structural damping conditions can be used for joint identification.

[0106] In summary, this invention establishes a motion equation related to the first-order modal response of nonlinear aerodynamic damping, derives an analytical displacement probability density function based on the Fokker–Planck–Kolmogorov equation, and constructs a self-supervised neural network embedded with probability density function information to jointly identify the nonlinear aerodynamic damping parameters and equivalent buffeting excitation intensity of flexible structures. This method has clear steps, well-defined physical constraints, and does not rely on aerodynamic damping parameter labels or a complete buffeting force spectrum, making it suitable for identifying nonlinear aerodynamic damping in wind tunnel tests and structural health monitoring data of flexible structures.

[0107] Example

[0108] This embodiment uses a bridge segment model tested in a wind tunnel as an example. Figure 3 This is a cross-sectional view of a bridge segment model used in a wind tunnel test. The model has a width B = 0.628 meters, a height D = 0.06 meters, a length L = 2.1 meters, and a mass per unit length m = 6.524. Moment of inertia I = 0.214 Its vertical frequency and damping ratio are as follows: Hz, Test wind angle of attack The turbulence intensity is I u =11%. Experimental procedure: ① A bridge segment model was built in the wind tunnel laboratory ( Figure 3 ); ② In the bridge segment model ( Figure 3 ) Place a laser displacement meter below; ③ For the bridge segment model ( Figure 3 Apply wind angle of attack and turbulence intensity I u =11% of the incoming turbulent flow; ④ In the bridge segment model ( Figure 3 Under vortex-induced resonance, the laser displacement gauge acquires the bridge segment model ( Figure 3 Vertical displacement at each measurement time.

[0109] Figure 4 for Figure 3 The wind tunnel test of a bridge segment model sampled in this application compared the displacement PDF, aerodynamic damping, and buffeting excitation intensity identified by the nonlinear aerodynamic damping self-supervised identification method for bridges with theoretical values. Figure 4 (a) For converted wind speed A comparison of the identified and target values ​​of the probability density function (PDF) of the bridge displacement response is used to demonstrate the preliminary fitting ability of the method of the present invention to the main peak and non-Gaussian tail characteristics of the response statistical distribution at low wind speeds. Figure 4 (b) For converted wind speed A comparison of the identified and target values ​​of the probability density function (PDF) of the bridge displacement response is used to illustrate the method's accurate capture performance of the tail probability of the hardened non-Gaussian response in the initial segment of the vortex-induced vibration locking interval. Figure 4 (c) is the converted wind speed A comparison of the identified and target values ​​of the probability density function (PDF) of the bridge displacement response is used to verify the method's ability to maintain the overall shape of the response statistical distribution within the locked interval. Figure 4 (d) is the converted wind speed A comparison of the identified and target values ​​of the probability density function (PDF) of the bridge displacement response is used to demonstrate the method's ability to stably reproduce the response probability distribution even at high wind speeds. Figure 4 (e) represents the buffeting excitation intensity under different equivalent wind speeds. The comparison chart of the identified value and the target value is used to verify the accuracy of the joint inversion of the buffeting force intensity under the condition of unknown wind load spectrum. Figure 4 (f) represents the nonlinear aerodynamic damping ratio under different equivalent wind speeds. The comparison chart of the identified values ​​and target values ​​is used to illustrate the accuracy of the method in retrieving the variation law of amplitude-dependent aerodynamic damping parameters with displacement amplitude.

[0110] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A self-supervised identification method for nonlinear aerodynamic damping of long-span bridges based on hardened non-Gaussian response probability density constraints, characterized in that... Includes the following steps: S1. Obtain the vertical displacement response data of the structural components under crosswind excitation in the crosswind direction, which is used to represent the generalized displacement response corresponding to the first-order mode of the structure. S2. Preprocess the vertical displacement response data from step S1 to obtain a stationary random displacement response sequence for parameter identification; divide the stationary random displacement response sequence into several data segments and obtain the target displacement probability density function using a density estimation method; simultaneously, use multiple structural damping conditions for joint identification, and construct the target displacement probability density function for the i-th condition respectively. ; S3. Establish the equations of motion related to the first-order modal response, which include nonlinear aerodynamic damping. The equations of motion related to the first-order modal response are expressed as follows: in, Indicates generalized structural mass; Indicates generalized displacement; Indicates the inherent structural damping ratio; Represents the natural circular frequency corresponding to the fundamental mode of the structure; This represents the first generalized force, which is composed of the flutter force. and self-excitation The sum of them constitutes, that is ; S4. Substitute the van der Pol transform into the motion equations related to the first-order modal response in step S3 to obtain the stochastic differential equations controlling the amplitude and phase processes. Use the Wong–Zakai correction to transform the stochastic differential equations into Itô form. Perform periodic averaging over one vibration period to obtain the average Itô stochastic differential equation for the amplitude process. Based on this, establish the one-dimensional Fokker–Planck–Kolmogorov equation. Under zero-probability boundary conditions, solve the Fokker–Planck–Kolmogorov equation to obtain the amplitude probability density function. Based on the amplitude probability density function... The analytical displacement probability density function of the displacement response is obtained through integral transformation; S5. Output parameters of the constructed PDF-ThetaNet Substituting the analytical displacement probability density function obtained in step S4, we obtain the predicted displacement probability density function. and the target displacement probability density function obtained in step S2 By comparison, self-supervised identification under the condition of no aerodynamic damping parameter labeling is achieved; S6. To simultaneously constrain the principal probability density region and harden the non-Gaussian tail region, a composite probability distribution loss function is constructed, and the PDF-ThetaNet constructed in step S5 is trained. S7. Substitute the parameter vector obtained after training in step S6 into the nonlinear aerodynamic damping model to output the amplitude-dependent nonlinear aerodynamic damping parameters and flutter excitation intensity of the long-span bridge structure.

2. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 1, characterized in that, In step S1, the structural components are the main beams of long-span bridges, bridge towers, or other wind-sensitive flexible components subjected to crosswinds; the vertical displacement response data are obtained from wind tunnel tests, on-site structural health monitoring systems, or numerical simulations.

3. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 1, characterized in that, Step S1 involves dimensionless processing of the vertical displacement response data to eliminate the influence of structural dimensions on subsequent aerodynamic damping identification. ,in, Represented as dimensionless displacement, This is represented by the vertical displacement response data of a structural component under crosswind excitation, i.e., generalized displacement. This represents the characteristic width of the main beam of the bridge or the structure to be identified.

4. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 1, characterized in that, Step S2: Vertical displacement response data preprocessing methods include mean removal and outlier elimination; density estimation methods include kernel density estimation, histogram normalization, or other probability density estimation.

5. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 1, characterized in that, Self-excitation force in step S3 Nonlinear aerodynamic damping Control, the self-excited force is represented as: in, This represents the nonlinear aerodynamic damping ratio related to the displacement amplitude. Indicates generalized structural mass; Represents the natural circular frequency corresponding to the fundamental mode of the structure; nonlinear aerodynamic damping ratio. Represented as a polynomial function of dimensionless displacement: in, Indicates air density, Indicates the width of the structure; Indicates the effective mass per unit height or unit length. Indicates the coefficients of the aerodynamic damping model to be identified; Flutter force In practical engineering applications, this is usually unknown; therefore, the equivalent stochastic excitation under the unit generalized mass is defined as: And Modeled as a Gaussian white noise process in the Stratoovich sense, its one-sided power spectral density at the fundamental frequency of the structure is denoted as... .

6. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 5, characterized in that, In step S4, since the structural response exhibits a narrow-band random process within the locked region of vortex-induced vibration, the following van der Pol transform is introduced: in, and For generalized harmonic functions, , ,and It is a stochastic process, where Represents the phase process, Indicates the amplitude process; Substituting the van der Pol transform into the motion equations related to the first-order modal response in step S3, we obtain the stochastic differential equations controlling the amplitude and phase processes: in, and The coefficients represent the drift terms, indicating the effects of structural damping and nonlinear aerodynamic damping on amplitude and phase, respectively. and The diffusion term coefficient is used to characterize the random disturbance caused by the buffeting force; because Modeled as Gaussian white noise in the Stratoovich sense, the above stochastic differential equation is transformed into Itô form using the Wong–Zakai correction: in, This represents a standard Wiener procedure. and This represents the corrected drift term. and Indicates the diffusion coefficient; By periodically averaging over a vibration period, the average Itô stochastic differential equation for the amplitude process is obtained, and based on this, a one-dimensional Fokker–Planck–Kolmogorov equation is established; under stationary conditions, the amplitude probability density function satisfies: in, Represents the amplitude probability density function. This represents the amplitude drift term after periodic averaging. This represents the diffusion term after periodic averaging; Under the zero-probability boundary condition, solving the above Fokker–Planck–Kolmogorov equation yields the amplitude probability density function: in, This is the normalization constant; Based on the amplitude probability density function The analytical probability density function of the displacement response is obtained through integral transform: in, It represents the natural circular frequency corresponding to the basic mode of the structure.

7. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 6, characterized in that, In step S5, a PDF-ThetaNet neural network is constructed to output the parameter vector to be identified. ,in, This represents the nonlinear aerodynamic damping parameters obtained by the neural network. This represents the equivalent chattering excitation intensity obtained by the neural network; PDF-ThetaNet includes an initial parameter input layer, an input projection layer, a residual multilayer perceptron module, an output projection layer, and a parameter mapping layer; the initial parameter input layer is used to input the initial guessed parameter vector. The input projection layer will The model is mapped to a high-dimensional hidden feature space; a residual multilayer perceptron module is used to extract parameter features, which consists of layer normalization, SiLU activation function, and fully connected layers; an output projection layer is used to output the estimated parameter vector to obtain the final parameters to be identified; the parameters output by PDF-ThetaNet are then processed. Substituting the analytical displacement probability density function from step S4, we obtain the predicted displacement probability density function. By comparison The result obtained in step S2 This enables self-supervised identification under conditions without aerodynamic damping parameter labels.

8. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 7, characterized in that, In step S6, a composite loss function is constructed to simultaneously constrain the principal probability density region and harden the non-Gaussian tail region: in, , and Let these represent the logarithmic probability density loss, the original probability density loss, and the cumulative distribution function loss, respectively, with the corresponding weighting coefficients set to... , , and ; The logarithmic probability density loss is used to enhance the contribution of the low-probability tail region in the optimization process, and is suitable for capturing the large-amplitude tail features in hardened non-Gaussian responses; the original probability density loss is used to constrain the main probability density region, preventing the network from focusing only on the tail and ignoring the main peak region of the probability density; the cumulative distribution function loss is used to constrain the cumulative probability consistency between the predicted probability distribution and the target probability distribution, thereby improving the overall probability distribution fitting stability.

9. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 8, characterized in that, Step S6 uses the Adam optimizer for initial optimization of PDF-ThetaNet, followed by local fine-tuning using the L-BFGS optimizer to obtain the final converged parameters to be identified. .

10. The self-supervised identification method for nonlinear aerodynamic damping of long-span bridges as described in claim 9, characterized in that, In step S7, the results obtained after training in step S6 are used... Substituting into the nonlinear aerodynamic damping model, we obtain the amplitude-dependent nonlinear aerodynamic damping curve of the structure: Simultaneously output equivalent chattering excitation intensity .

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