Method and device for determining aerodynamic damping of bridge tower based on unscented Kalman filter

Abstract

The invention relates to the technical field of wind resistance design of bridge structures, discloses a bridge tower aerodynamic damping determination method and device based on unscented Kalman filtering, and aims to solve the technical problem that existing aerodynamic damping identification is not accurate enough. The scheme mainly comprises the following steps: determining an initial nonlinear aerodynamic damping model comprising a plurality of unknown model parameters of the bridge tower, and constructing a state equation comprising the model parameters; obtaining a vortex-induced vibration response time history of the bridge tower model by using a wind tunnel test, and constructing an observation equation; inputting the state equation and the observation equation into an unscented Kalman filter, and determining values of model parameters by using the unscented Kalman filter; and substituting the values of the model parameters into the initial nonlinear aerodynamic damping model to obtain a nonlinear aerodynamic damping model, and determining aerodynamic damping by using the nonlinear aerodynamic damping model. Accurate identification of aerodynamic damping is achieved, the requirement for the actual measurement response time history is lowered, and the method is particularly suitable for accurately evaluating the wind-induced response of the structure.

Description

Technical Field

[0001] The present application relates to the technical field of wind-resistant design of bridge structures, and in particular to a method and device for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering. Background Art

[0002] As the span increases, the bridge structure (whether the main span or the bridge tower) will become more flexible and more wind-sensitive. Therefore, accurately evaluating the wind-induced vibration response of the bridge tower is of great significance for the design of high-flexibility bridge tower wind resistance.

[0003] Vortex-induced vibration is a wind-induced vibration phenomenon that occurs in long-span bridges at low wind speeds. Although vortex-induced vibration is not as destructive as divergent vibration, it is prone to occur at low wind speeds and has a significant impact on the safety and comfort of the structure during operation. For tall structures such as bridge towers, aerodynamic damping is a key parameter for evaluating vortex-induced vibration. Pneumatic positive damping is beneficial for structural vibration control, reducing wind-induced response, and increasing structural safety and comfort; pneumatic negative damping will have a negative effect of exacerbating structural response, and have an adverse effect on the comfort and safety of the structure. Accurate identification of aerodynamic damping is the first step in accurately evaluating the wind-induced response of the structure.

[0004] The mechanism of aerodynamic damping is very complex, and it is currently impossible to analyze the problem of aerodynamic damping through a completely analytical method. In engineering, the existing research methods for aerodynamic damping are mainly wind tunnel test methods combined with numerical analysis, that is, by making a scaled aeroelastic model of the structure, collecting information such as displacement, velocity, acceleration, etc. through wind tunnel tests, and identifying aerodynamic damping based on system modal parameter identification methods (such as random decrement method, aerodynamic damping identification based on Hilbert transform method, aerodynamic damping identification based on PDF-fitting, etc.) to guide design research.

[0005] The existing technology has certain limitations when identifying amplitude-dependent aerodynamic damping, resulting in inaccurate identification of aerodynamic damping. For example, the aerodynamic damping identification method based on PDF-fitting can only identify amplitude-dependent aerodynamic damping for measured data containing obvious noise of a single-degree-of-freedom system. When studying amplitude-dependent aerodynamic damping, the aerodynamic damping identification method based on Hilbert transform (HT) can only conduct amplitude-dependent aerodynamic damping identification research in measured data starting from static vibration to stable amplitude. Summary of the invention

[0006] The present application aims to solve the technical problem that the existing aerodynamic damping identification method has certain limitations, resulting in inaccurate identification of aerodynamic damping, and proposes a method and device for determining the aerodynamic damping of a bridge tower based on unscented Kalman filtering.

[0007] The technical solution adopted by this application to solve the above technical problems is:

[0008] In a first aspect, the present application provides a method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering, the method comprising:

[0009] Determine an initial nonlinear aerodynamic damping model of the bridge tower including a plurality of unknown model parameters, and construct a state equation including the model parameters according to the initial nonlinear aerodynamic damping model;

[0010] The vortex-induced vibration response time history of the bridge tower model is obtained by using a wind tunnel test, and an observation equation is constructed according to the vortex-induced vibration response time history;

[0011] Inputting the state equation and the observation equation into an unscented Kalman filter, and using the unscented Kalman filter to determine the values ​​of model parameters;

[0012] The values ​​of the model parameters are substituted into the initial nonlinear aerodynamic damping model to obtain the nonlinear aerodynamic damping model, and the aerodynamic damping is determined using the nonlinear aerodynamic damping model.

[0013] Furthermore, the initial nonlinear aerodynamic damping model is as follows:

[0014]

[0015] Among them, ξ a represents the time-varying dimensionless displacement nonlinear aerodynamic damping, ρ represents the air density, B represents the top width of the model, m s represents the effective mass per unit length, y represents the vibration displacement, B 1 , B 2 , B 3 and B 4 Represents model parameters.

[0016] Furthermore, a state equation containing model parameters is constructed according to the initial nonlinear aerodynamic damping model, specifically including:

[0017] An initial state equation is constructed, the initial nonlinear aerodynamic damping model is substituted into the initial state equation and then derived, and the initial state equation is rewritten according to the crosswind response motion equation of the tall structure to obtain a rewritten state equation.

[0018] Furthermore, the initial state equation is as follows:

[0019]

[0020] The equation of motion for the crosswind response of the tall structure is as follows:

[0021]

[0022] The rewritten state equation is as follows:

[0023]

[0024] Where X represents the state vector, represents the first derivative of y, i.e. the vibration velocity, ω s represents the natural frequency of the model, ξ s represents the system damping, S represents the structure under wind load, T represents the matrix transpose, represents the second-order derivative of y, i.e., vibration acceleration, m represents the model mass, Q x (t) represents the fundamental modal force that is only related to time, M s Represents the mass of the system involved in the vibration.

[0025] Furthermore, an observation equation is constructed according to the vortex-induced vibration response time history, specifically including:

[0026] The vibration displacement and vibration velocity in the vortex-induced vibration response time history are used as observation variables to construct an observation equation, which is as follows:

[0027]

[0028] Where Z represents the observation vector.

[0029] Further, using the unscented Kalman filter to determine the value of the model parameter specifically includes:

[0030] Initial values ​​are set for the model parameters in the state equation, and the unscented Kalman filter is iteratively updated based on the initial values ​​to obtain corresponding predicted values. When the predicted values ​​match the observed values ​​corresponding to the observation equation, the iteration is stopped to obtain the values ​​of the model parameters in the state equation at this time.

[0031] Further, the unscented Kalman filter performs cyclic update iteration based on the initial value, specifically including:

[0032] For the state vector and covariance matrix, a set of sigma points are generated according to the unscented transformation;

[0033] The generated sigma point is transformed nonlinearly through the state transfer function to obtain the predicted sigma point, and the mean and covariance of the predicted state vector are calculated based on the predicted sigma point;

[0034] The predicted sigma point is transformed nonlinearly through the measurement function to obtain the predicted measured sigma point, and the mean and covariance of the measurement vector are calculated based on the predicted measured sigma point;

[0035] Calculate the Kalman gain and update the state vector and covariance matrix according to the calculation results;

[0036] Repeat the above steps until the measurement vector converges to the observed value.

[0037] Furthermore, the method further comprises:

[0038] The aerodynamic damping under different amplitudes is calculated based on the nonlinear aerodynamic damping model, and a relationship curve between amplitude and aerodynamic damping is drawn according to the calculation results. The calculation formula is as follows:

[0039]

[0040] Among them, ξ a1 represents the time-varying dimensionless amplitude nonlinear aerodynamic damping, represents the first-order derivative of y, that is, the vibration speed, ω represents the angular frequency, t represents the vibration time, y max Indicates amplitude.

[0041] In a second aspect, the present application provides a device for determining aerodynamic damping of a bridge tower based on an unscented Kalman filter, characterized in that the device is used to implement the steps of the method for determining aerodynamic damping of a bridge tower based on an unscented Kalman filter as described in the first aspect.

[0042] The beneficial effects of the present application are as follows: the method and device for determining the aerodynamic damping of a bridge tower based on unscented Kalman filtering provided by the present application determine the model parameters in the nonlinear aerodynamic damping model based on the unscented Kalman filtering technology, and identify the amplitude-dependent aerodynamic damping based on the nonlinear aerodynamic damping model with the determined model parameters. This not only realizes the accurate identification of the aerodynamic damping, but also reduces the requirements for the measured response time history, and is suitable for bridge segments with different cross-sectional models and tall structures with rectangular and cylindrical cross-sections. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 A schematic flow chart of a method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering provided in an embodiment of the present application;

[0044] Figure 2 An iterative schematic diagram of an unscented Kalman filter provided in an embodiment of the present application;

[0045] Figure 3 A curve diagram of matching a predicted value of a vibration displacement with a measured value provided in an embodiment of the present application;

[0046] Figure 4 A curve diagram of matching a predicted value of a vibration velocity with an actual measured value provided in an embodiment of the present application;

[0047] Figure 5 A schematic diagram of an amplitude-aerodynamic damping curve corresponding to different moments in an iterative process of a UKF provided in an embodiment of the present application;

[0048] Figure 6 A schematic diagram of an amplitude-aerodynamic damping curve corresponding to setting different initial values ​​for a space state vector provided in an embodiment of the present application;

[0049] Figure 7 A schematic diagram of an amplitude-aerodynamic damping curve drawn according to a nonlinear aerodynamic damping model provided in an embodiment of the present application. DETAILED DESCRIPTION

[0050] In order to enable those skilled in the art to better understand the solution of the present application, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application.

[0051] The technical solution of the embodiment of the present application is applicable to application scenarios where accurate identification of aerodynamic damping is required to accurately evaluate the wind-induced response of the structure. Since the current aerodynamic damping identification method has certain limitations on the measured data, it leads to the problem of poor accuracy. Based on this, the technical solution of the present application is proposed. In the embodiment of the present application, an initial nonlinear aerodynamic damping model of the bridge tower including multiple unknown model parameters is determined, and a state equation containing model parameters is constructed according to the initial nonlinear aerodynamic damping model; the vortex-induced vibration response time history of the bridge tower model is obtained by wind tunnel testing, and an observation equation is constructed according to the vortex-induced vibration response time history; the state equation and the observation equation are input into an unscented Kalman filter, and the value of the model parameter is determined by using the unscented Kalman filter; the value of the model parameter is substituted into the initial nonlinear aerodynamic damping model to obtain a nonlinear aerodynamic damping model, and the aerodynamic damping is determined by using the nonlinear aerodynamic damping model.

[0052] Unscented Kalman Filter (UKF) is a nonlinear Kalman filter algorithm that combines the unscented transform (UT) and the standard Kalman filter system to deal with the state estimation problem of nonlinear systems. The basic principle of unscented Kalman filtering is to linearize the nonlinear system equations through unscented transformation. It uses a set of weighted sampling points (Sigma points) to approximate the probability distribution of nonlinear functions, and calculates the mean and covariance of nonlinear functions through these sampling points. This method is more accurate than the traditional Taylor series expansion (such as the extended Kalman filter EKF) because it considers the expansion of random variables rather than just first-order linearization. The advantages of unscented Kalman filtering include high estimation accuracy and applicability to nonlinear systems. It performs well in dealing with state estimation of nonlinear systems, especially when the system model is highly nonlinear and noisy.

[0053] The embodiment of the present application introduces the nonlinear aerodynamic damping used in the existing research on the crosswind response of high-rise building structures as an improved nonlinear aerodynamic damping model of bridge towers, and through a free vibration wind tunnel experiment of the aerodynamic elastic model of the bridge tower, the time history of the bridge model when vortex-induced vibration occurs is collected, and then the unknown model parameters in the nonlinear aerodynamic damping model are determined based on the unscented Kalman filter. Specifically, a state equation is created according to the model parameters, and an observation equation is created according to the time history measured by the model test. Then, the state equation and the observation equation are input into the unscented Kalman filter for cyclic iterative update until convergence, so as to determine the value of the model parameter, and the determined value of the model parameter is substituted into the nonlinear aerodynamic damping model. The nonlinear aerodynamic damping model can be used to identify the aerodynamic damping, thereby realizing accurate identification of the aerodynamic damping and reducing the requirements for the measured response time history.

[0054] The technical solutions in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all of the embodiments.

[0055] See also Figure 1 The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering provided in an embodiment of the present application comprises the following steps:

[0056] S1. Determine an initial nonlinear aerodynamic damping model of a bridge tower including a plurality of unknown model parameters, and construct a state equation including the model parameters according to the initial nonlinear aerodynamic damping model.

[0057] The embodiment of the present application introduces the nonlinear aerodynamic damping used in the prior art to study the crosswind response of high-rise building structures as an improved bridge tower nonlinear aerodynamic damping model, which includes three:

[0058] (1) The nonlinear aerodynamic damping is expressed as a polynomial function model of time-varying displacement, as follows:

[0059]

[0060] (2) The nonlinear aerodynamic damping is expressed as a polynomial function model with time-varying amplitude, as follows:

[0061]

[0062] (3) The nonlinear aerodynamic damping is expressed as a polynomial function model of time-varying velocity, as follows:

[0063]

[0064] Among them, ξ a represents the time-varying dimensionless displacement nonlinear aerodynamic damping, ξ a1 represents the time-varying dimensionless amplitude nonlinear aerodynamic damping, ξ a2 represents the time-varying dimensionless velocity nonlinear aerodynamic damping, ρ represents the air density, B represents the model top width, m s represents the effective mass per unit length, y represents the time-varying dimensionless displacement, i.e., the vibration displacement, y max represents the time-varying dimensionless amplitude, i.e., the amplitude, represents the first-order derivative of y, which is the instantaneous dimensionless velocity, that is, the vibration velocity, A 1 , A 2 , A 3 and A 4 and B 1 , B 2 , B 3 and B 4 represents model parameters, U represents wind speed, and K represents dimensionless frequency.

[0065] The relationship between the polynomial function model of the time-varying displacement and the time-varying amplitude is established based on the harmonic balance method, and the relationship between the polynomial function model of the time-varying displacement and the time-varying velocity is established by derivatizing the time-varying displacement.

[0066] For the harmonic balance method: two aerodynamic damping models can be considered in simple harmonic oscillation y = ysin (ωt) max The work done in one vibration cycle is equal, that is:

[0067]

[0068] We can get:

[0069]

[0070] The relationship between the two nonlinear aerodynamic damping of time-varying displacement and time-varying amplitude is established, where ω represents the angular frequency and t represents the vibration time.

[0071] The embodiment of the present application uses a polynomial function model of time-varying displacement as the initial nonlinear aerodynamic damping model, where the unknown model parameters are B 1 , B 2 , B 3 and B 4 .

[0072] In an embodiment of the present application, a state equation including model parameters is constructed according to the initial nonlinear aerodynamic damping model, specifically including:

[0073] An initial state equation is constructed, the initial nonlinear aerodynamic damping model is substituted into the initial state equation and then derived, and the initial state equation is rewritten according to the crosswind response motion equation of the tall structure to obtain a rewritten state equation.

[0074] Specifically, before using the unscented Kalman filter to determine the model parameters, the displacement state equation of the single-degree-of-freedom system needs to be defined. For the structure under wind load, considering its wind load loading process, it can be regarded as a zero-mean Gaussian white noise input. Then the initial state equation including the structural frequency, system damping and external load process can be obtained:

[0075]

[0076] Substitute the initial nonlinear aerodynamic damping model into the initial state equation and then take the derivative, and then calculate the motion equation according to the crosswind response of the tall structure:

[0077]

[0078] The rewritten state equation can be obtained:

[0079]

[0080] Where X represents the state vector, represents the first derivative of y, i.e. the vibration velocity, ω s represents the natural frequency of the model, ξ s represents the system damping, S represents the structure under wind load. Considering its wind load loading process, it can be regarded as a zero-mean Gaussian white noise input, T represents matrix transposition, represents the second-order derivative of y, i.e., vibration acceleration, m represents the model mass, Q x (t) represents the fundamental modal force that is only related to time, M s Represents the mass of the system involved in the vibration.

[0081] In the rewritten state equation of the embodiment of the present application, the dimension of the augmented state space variable is 7, which includes high-order terms of the polynomial function used to identify the system damping.

[0082] S2. Obtain the vortex-induced vibration response time history of the bridge tower model by using a wind tunnel test, and construct an observation equation according to the vortex-induced vibration response time history.

[0083] Wind tunnel test is an important experimental method used to simulate fluid dynamic phenomena in actual environments. The embodiment of the present application tests the bridge tower model in a wind tunnel, records and analyzes the vibration response data of the model under different wind speed conditions, and then obtains the vortex-induced vibration response time history of the bridge tower model. Specifically, in the vortex-induced vibration study of the bridge tower model, the vibration of the bridge tower under different natural wind conditions can be simulated by controlling the wind speed and wind direction in the wind tunnel. Through the sensors installed on the model, the vibration response data of the bridge tower model can be recorded in real time, including parameters such as displacement, velocity, acceleration, etc. The changes of these data over time constitute the vortex-induced vibration response time history.

[0084] In practical applications, it is first necessary to make a scaled model based on the geometry and physical characteristics of the actual bridge tower to ensure that the model can accurately reflect the fluid dynamic characteristics of the prototype. Then the model is placed in a wind tunnel to simulate different natural wind conditions by adjusting the wind speed and direction. Then, sensors are used to measure the vibration response of the model, such as displacement, acceleration, etc., and these data will be recorded in real time. Finally, the collected data is processed and analyzed to extract the characteristics of vortex-induced vibration, such as amplitude and frequency, and then the vortex-induced vibration response time history is obtained.

[0085] After obtaining the vortex-induced vibration response time history of the bridge tower model, the embodiment of the present application constructs an observation equation according to the vortex-induced vibration response time history, specifically including:

[0086] The vibration displacement and vibration velocity in the vortex-induced vibration response time history are used as observation variables to construct the observation equation, that is, the measured vibration displacement signal and velocity signal (the velocity signal can also be obtained by derivation of the vibration displacement signal) are selected as the observation equation of the unscented Kalman filter. The observation equation is as follows:

[0087]

[0088] Where Z represents the observation vector.

[0089] S3. Input the state equation and the observation equation into an unscented Kalman filter, and use the unscented Kalman filter to determine the values ​​of model parameters.

[0090] After creating the state equation and observation equation, the state equation and observation equation are simultaneously input into the unscented Kalman filter, and the filter is updated and iterated. Among them, the state equation is the system state equation containing the parameters of interest, i.e., the model parameters, and the observation equation contains the measured data. For a system containing unknown variables, when there is a set of observation data (such as the measured response signal), an initial value is arbitrarily given to the system, so that the initial value is updated and iterated through UKF to obtain the predicted value. When the updated and iterated predicted value completely matches the measured value of the observation data, the parameters obtained from the prediction will be consistent with the corresponding parameters in the actual system. The identification process of the UKF method mainly includes three parts: time update, prediction update, and state update. Through such a cyclic iteration, when the variance is minimized, the model parameters of interest can be accurately identified.

[0091] In an embodiment of the present application, the value of the model parameter is determined by using the unscented Kalman filter, specifically including:

[0092] Initial values ​​are set for the model parameters in the state equation, and the unscented Kalman filter is iteratively updated based on the initial values ​​to obtain corresponding predicted values. When the predicted values ​​match the observed values ​​corresponding to the observation equation, the iteration is stopped to obtain the values ​​of the model parameters in the state equation at this time.

[0093] The specific steps of performing cyclic update iteration of the unscented Kalman filter based on the initial value include:

[0094] S31, for the state vector and the covariance matrix, generate a set of sigma points according to the unscented transformation;

[0095] S32, performing nonlinear transformation on the generated sigma point through a state transfer function to obtain a predicted sigma point, and calculating the mean and covariance of the predicted state vector according to the predicted sigma point;

[0096] S33, performing nonlinear transformation on the predicted sigma point through the measurement function to obtain a predicted measurement sigma point, and calculating the mean and covariance of the measurement vector according to the predicted measurement sigma point;

[0097] S34, calculating the Kalman gain and updating the state vector and the covariance matrix according to the calculation result;

[0098] S35. Repeat the above steps until the measurement vector converges to the observed value.

[0099] Please refer to the figure Figure 2 In practical applications, the state equation X and observation equation Z of nonlinear dynamic systems can be described by discrete time equations as follows:

[0100] X k+1 =F(Xk )+w k ;

[0101] Z k =H(X k )+v k ;

[0102] In the formula, F() and H() represent the functions of state and observation respectively, k represents the instantaneous time index, which can be understood as the iteration proceeds to the kth step, w k =w(kΔt) means the covariance matrix The zero-mean process Gaussian noise vector, v k =v(kΔt) means the covariance matrix is the zero-mean measurement Gaussian noise vector, Δt represents the time step, and E[] represents the expectation operator.

[0103] The implementation of unscented Kalman filtering mainly includes the following steps:

[0104] Step A: Initialize state estimate X 0 and the covariance matrix P 0 :

[0105]

[0106]

[0107] Step B: Calculate the sigma point, that is, according to X 0 and P 0 Generate sigma point χ k-1,i , as follows:

[0108] At time step k-1, a scaled unscented transform is used to generate (2L+1) sample sigma points, where L is the dimension of the state vector X. In the embodiment of the present application, L=7, as shown below:

[0109]

[0110] λ=α 2 (L+κ)-L;

[0111] Among them, χ k-1,i represents the sigma point value of the i-th row in the k-1th iteration. When i=0, it corresponds to the initial value of the sample sigma point in the first iteration. λ represents the first scaling parameter, and α determines The sigma point distribution around the iteration value at the k-1th step is usually set to a small positive value (such as 1e -3 ), κ represents the second scaling parameter, which is usually set to 0, represents the i-th row of the matrix square root.

[0112] Step C: Time update, that is, according to the process noise covariance Q k The predicted state vector obtained and its covariance The details are as follows:

[0113] State Vector With its covariance It can be predicted as:

[0114] χ k,i =F(χ k-1,i ), (i=0,1,…,2L);

[0115]

[0116]

[0117] Among them, W i (m) Represents the distribution weight coefficient of the desired state vector iteration, W i (c) represents the allocation weight coefficient of the covariance state vector, β represents the parameter of the residual higher-order effects in the covariance estimation, and β=2 is the optimal value of the Gaussian distribution.

[0118] Step D: update the prediction, i.e., based on the measurement noise covariance R k Get the predicted measurement vector and covariance And calculate the covariance matrix The details are as follows:

[0119] Measuring vector With its covariance It can be predicted as:

[0120] Z k,i =H(χ k,i ), (i=1,2,…,2L);

[0121]

[0122] Covariance matrix It is expressed as:

[0123]

[0124] Step E: state update, i.e., according to the Kalman gain K gain,k and the observation vector Z k Update the state vector and the covariance matrix P k , as follows:

[0125]

[0126]

[0127] Through UKF iteration, the value of the model parameters changes accordingly with the number of iterations. The measurement vector is updated after k iterations. It will converge to the observed value, and then obtain the state vector that has been updated after k iterations. The values ​​of the model parameters in are used to complete the determination of the values ​​of the model parameters.

[0128] In practical applications, the initial values ​​of state space variables can be set, for example:

[0129] Case 1: X 0 =diag[0 0 -3 -5 300 0.2 0];

[0130] Case 2: X 0 =diag[0 0 -2 -10 250 0.3 0];

[0131] The initial values ​​of the aerodynamic damping parameters are determined based on the fact that aerodynamic damping is negative at 0 amplitude and gradually decreases as the amplitude increases. According to certain rules, the maximum amplitude of each variable is estimated and the initial covariance matrix is ​​determined, for example:

[0132] P 0 =diag[1 1 10 -1 10 -1 1 10 -1 100];

[0133] The maximum amplitude of dimensionless displacement and dimensionless velocity is first order, so their covariance can be assumed to be 1. The initial covariance of other parameters is similarly considered. k and R k The same considerations should be made for , which are considered to be similar constant matrices. The noise is usually much smaller than the variable itself. For example, the following assumptions can be made for process noise and state noise:

[0134] Q=[0 0 10 -10 10 -8 10 -8 10 -8 10 -6.5 ];

[0135] R = [10 -7 10 -4 10 -2 ];

[0136] Other parameters can also be set, for example: α = 1e -3 , β=2 and k=0.

[0137] Based on the above data, UKF can be used to iterate until convergence to complete the determination of the values ​​of the model parameters.

[0138] In the process of UKF iteration, the predicted value of vibration displacement or vibration velocity is matched with the measured value. Figure 3 A curve diagram showing the matching of the predicted value of vibration displacement with the measured value is shown. Figure 4 A curve schematic diagram showing the matching of a predicted value of vibration velocity with an actual measured value is shown.

[0139] Figure 5 A schematic diagram of the amplitude-aerodynamic damping curve corresponding to different moments in the UKF iteration process is shown. A schematic diagram of the relationship between the amplitude and aerodynamic damping corresponding to different moments in the UKF iteration process, where the amplitude-aerodynamic damping curves corresponding to the model parameters at three different iteration times of t = 0s, 25s, 50s and 100s are shown. The results show that as the UKF prediction and estimation results are continuously updated, the estimated aerodynamic damping gradually approaches and converges to the target value (the target value is the parameter value obtained by taking the average value of the last stable segment).

[0140] Figure 6 A schematic diagram of the amplitude-aerodynamic damping curve corresponding to different initial values ​​of the space state vector is shown. The results show that in both cases, the aerodynamic damping ratio that varies with amplitude finally converges to the same value, and there is only a difference in the estimation results at small amplitudes.

[0141] S4. Substituting the values ​​of the model parameters into the initial nonlinear aerodynamic damping model to obtain a nonlinear aerodynamic damping model, and determining the aerodynamic damping using the nonlinear aerodynamic damping model.

[0142] Specifically, the determined values ​​of the model parameters are finally substituted into the nonlinear aerodynamic damping model, and the nonlinear aerodynamic damping model can be used to identify the aerodynamic damping, thereby achieving accurate identification of the aerodynamic damping.

[0143] In the embodiment of the present application, the aerodynamic damping at different amplitudes is also calculated based on the nonlinear aerodynamic damping model, and a relationship curve diagram between the amplitude and the aerodynamic damping is drawn according to the calculation results. The calculation formula of the aerodynamic damping at different amplitudes is established according to the relationship between the polynomial function model of the time-varying displacement and the time-varying amplitude, that is:

[0144]

[0145] For the amplitude-aerodynamic damping curves drawn based on the calculation results of the nonlinear aerodynamic damping model, please refer to Figure 7 .

[0146] In summary, the method for determining the aerodynamic damping of a bridge tower based on an unscented Kalman filter provided in the embodiment of the present application introduces the nonlinear aerodynamic damping used in the existing research on the crosswind response of high-rise building structures as an improved nonlinear aerodynamic damping model of a bridge tower, and collects the time history of the bridge model when vortex-induced vibration occurs through a free vibration wind tunnel experiment of the aeroelastic model of the bridge tower, and then determines the unknown model parameters in the nonlinear aerodynamic damping model based on the unscented Kalman filter, specifically, creating a state equation according to the model parameters, creating an observation equation according to the time history measured by the model test, and then inputting the state equation and the observation equation into the unscented Kalman filter for cyclic iterative update until convergence, thereby determining the value of the model parameter, and substituting the determined value of the model parameter into the nonlinear aerodynamic damping model, and then using the nonlinear aerodynamic damping model to identify the aerodynamic damping, thereby achieving accurate identification of the aerodynamic damping and reducing the requirements for the measured response time history.

[0147] Based on the above technical solution, this embodiment also proposes a device for determining bridge tower aerodynamic damping based on unscented Kalman filtering, characterized in that the device is used to implement the steps of the method for determining bridge tower aerodynamic damping based on unscented Kalman filtering described in the embodiment of the present application.

[0148] It can be understood that since the device for determining the aerodynamic damping of a bridge tower based on unscented Kalman filtering described in the embodiment of the present application is a device for implementing the method for determining the aerodynamic damping of a bridge tower based on unscented Kalman filtering described in the embodiment, for the device disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the partial description of the method, which will not be repeated here.

Claims

1. A method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering, characterized in that: The method comprises: Determine an initial nonlinear aerodynamic damping model of the bridge tower including a plurality of unknown model parameters, and construct a state equation including the model parameters according to the initial nonlinear aerodynamic damping model; The vortex-induced vibration response time history of the bridge tower model is obtained by using a wind tunnel test, and an observation equation is constructed according to the vortex-induced vibration response time history; Inputting the state equation and the observation equation into an unscented Kalman filter, and using the unscented Kalman filter to determine the values ​​of model parameters; The values ​​of the model parameters are substituted into the initial nonlinear aerodynamic damping model to obtain the nonlinear aerodynamic damping model, and the aerodynamic damping is determined using the nonlinear aerodynamic damping model.

2. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 1, characterized in that: The initial nonlinear aerodynamic damping model is as follows: Among them, ξ a represents the time-varying dimensionless displacement nonlinear aerodynamic damping, ρ represents the air density, B represents the top width of the model, m s represents the effective mass per unit length, y represents the vibration displacement, and B1, B2, B3 and B4 represent the model parameters.

3. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 2 is characterized in that: The state equation containing model parameters is constructed according to the initial nonlinear aerodynamic damping model, specifically including: An initial state equation is constructed, the initial nonlinear aerodynamic damping model is substituted into the initial state equation and then derived, and the initial state equation is rewritten according to the crosswind response motion equation of the tall structure to obtain a rewritten state equation.

4. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 3 is characterized in that: The initial state equation is as follows: The equation of motion for the crosswind response of the tall structure is as follows: The rewritten state equation is as follows: Where X represents the state vector, represents the first derivative of y, i.e. the vibration velocity, ω s represents the natural frequency of the model, ξ s represents the system damping, S represents the structure under wind load, T represents the matrix transpose, represents the second-order derivative of y, i.e., vibration acceleration, m represents the model mass, Q x (t) represents the fundamental modal force that is only related to time, M s Represents the mass of the system involved in the vibration.

5. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 4, characterized in that: The observation equation is constructed according to the vortex-induced vibration response time history, specifically including: The vibration displacement and vibration velocity in the vortex-induced vibration response time history are used as observation variables to construct an observation equation, which is as follows: Where Z represents the observation vector.

6. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 4, characterized in that: Determining the values ​​of model parameters using the unscented Kalman filter specifically includes: Initial values ​​are set for the model parameters in the state equation, and the unscented Kalman filter is iteratively updated based on the initial values ​​to obtain corresponding predicted values. When the predicted values ​​match the observed values ​​corresponding to the observation equation, the iteration is stopped to obtain the values ​​of the model parameters in the state equation at this time.

7. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 6, characterized in that: The unscented Kalman filter performs cyclic update iteration based on the initial value, specifically including: For the state vector and covariance matrix, a set of sigma points are generated according to the unscented transformation; The generated sigma point is transformed nonlinearly through the state transfer function to obtain the predicted sigma point, and the mean and covariance of the predicted state vector are calculated based on the predicted sigma point; The predicted sigma point is transformed nonlinearly through the measurement function to obtain the predicted measured sigma point, and the mean and covariance of the measurement vector are calculated based on the predicted measured sigma point; Calculate the Kalman gain and update the state vector and covariance matrix according to the calculation results; Repeat the above steps until the measurement vector converges to the observed value.

8. The method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering according to claim 2, characterized in that: The method further comprises: The aerodynamic damping under different amplitudes is calculated based on the nonlinear aerodynamic damping model, and a relationship curve between amplitude and aerodynamic damping is drawn according to the calculation results. The calculation formula is as follows: Among them, ξ a1 represents the time-varying dimensionless amplitude nonlinear aerodynamic damping, represents the first-order derivative of y, that is, the vibration speed, ω represents the angular frequency, t represents the vibration time, y max Indicates amplitude.

9. A device for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering, characterized in that: The device is used to implement the steps of the method for determining aerodynamic damping of a bridge tower based on unscented Kalman filtering as described in any one of claims 1 to 8.

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