Multi-stable vortex-induced vibration aerodynamic damping estimation method, system and storage medium based on pdf analysis solution

Abstract

This invention discloses a method for estimating aerodynamic damping of multistable vortex-induced vibrations based on PDF analytical solutions. The method includes: constructing a polynomial aerodynamic damping model; establishing an analytical solution for the amplitude probability density distribution function, which is a function containing structural damping, buffeting force, external load, and aerodynamic damping coefficients; acquiring two sets of amplitude probability density distribution data under different structural damping conditions for the same vibration steady state; simultaneously fitting the two sets of distribution data using the analytical solution to identify the aerodynamic damping model coefficients and external loads; and obtaining the nonlinear aerodynamic damping estimate based on the obtained coefficients. This invention also discloses a system and storage medium for estimating aerodynamic damping of multistable vortex-induced vibrations based on PDF analytical solutions. This invention overcomes the problem of non-unique parameter identification from a single data source through joint inversion under dual damping conditions, and can accurately and efficiently identify strongly nonlinear and non-monotonic aerodynamic damping in multistable vortex-induced vibrations, providing a reliable means for wind-induced vibration safety assessment and wind-resistant design of engineering structures.

Description

Technical Field

[0001] This invention belongs to the field of structural wind engineering and aerodynamic analysis technology, specifically a method, system and storage medium for estimating multistable vortex-induced aerodynamic damping based on PDF analytical solutions. Background Technology

[0002] For flexible structures such as long-span bridges and high-rise buildings, as well as cylindrical cross-sections such as wind turbine towers and power transmission cables, vortex-induced vibrations with multiple stable amplitudes may occur. Structures starting with different initial amplitudes may reach different steady states, and the steady state depends on the vibration's development path. Multistable vortex-induced vibrations are related to the non-monotonicity of aerodynamic damping with respect to amplitude. The amplitude dependence of aerodynamic damping is related to complex fluid-structure interaction phenomena and flow separation phenomena in the surrounding flow field. Therefore, accurate estimation of aerodynamic damping has become a core challenge in assessing the safety and comfort of structures under wind-induced vibration.

[0003] Traditional methods for estimating aerodynamic damping primarily rely on wind tunnel testing and system identification techniques. For aerodynamic damping with weak nonlinearity and a monotonic function of amplitude, relatively simple and widely applicable methods such as least squares or stochastic decrement techniques are commonly used. However, for vortex-induced vibrations with multiple stable states, aerodynamic damping exhibits strong nonlinearity and is a non-monotonic function of amplitude. Currently, there is a lack of understanding of the relationship between multiple stable amplitudes and aerodynamic damping, as well as a lack of methods for identifying non-monotonic aerodynamic damping in multi-steady vortex-induced vibrations. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a method, system and storage medium for estimating multistable vortex-induced aerodynamic damping based on PDF analytical solutions, which aims to efficiently and accurately identify aerodynamic damping with strong nonlinear characteristics in hysteresis phenomena with multiple stable states.

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

[0006] This invention first proposes a method for estimating the aerodynamic damping of multistable vortex vibrations based on PDF analytical solutions, comprising the following steps:

[0007] Step 1: Based on the multiple stable states in multi-steady vortex-induced vibration, construct a polynomial aerodynamic damping model with structural amplitude as the variable;

[0008] Step 2: Based on the analytical solution of random wind vibration, establish the analytical solution of the amplitude probability density distribution function. The analytical solution of the amplitude probability density distribution function is a function that includes the coefficients of structural damping, buffeting force external load, and the polynomial aerodynamic damping model.

[0009] Step 3: For the same vibration steady state, obtain the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values ​​respectively;

[0010] Step 4: In the same vibration steady state, the analytical solution of the amplitude probability density distribution function established in Step 2 is used to simultaneously fit the two sets of amplitude probability density distribution data obtained in Step 3, so as to simultaneously identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

[0011] Step 5: Based on the coefficients of the identified polynomial aerodynamic damping model, obtain the estimated value of nonlinear aerodynamic damping for each steady state. Place the aerodynamic damping identification results corresponding to the amplitude under different steady states into the same horizontal coordinate system to obtain the identification results for multiple steady states.

[0012] Furthermore, in step one, the polynomial aerodynamic damping model is expressed as:

[0013]

[0014] in: Nonlinear aerodynamic damping; air density; The width of the structure; Indicates the effective structural mass per unit length; The amplitude of the structural vibration; These are the polynomial coefficients of the polynomial aerodynamic damping model, and all of them are constants; Let be the order of the polynomial.

[0015] Furthermore, in step two, the analytical solution of the amplitude probability density distribution function is expressed as:

[0016]

[0017] in: This represents the amplitude probability density distribution function; yes The reduction constants of make such that The integral is 1; It is the natural circular frequency; It is a buffeting force external load; It is structural damping.

[0018] Furthermore, in step three, the damping values ​​of the first group of structures and the damping values ​​of the second group of structures are different values.

[0019] Furthermore, in step four, the least squares method is used for fitting to identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

[0020] This invention also proposes a multistable vortex aerodynamic damping estimation system based on PDF analytical solutions, used to implement the multistable vortex aerodynamic damping estimation method based on PDF analytical solutions as described above, including:

[0021] The model building module is used to build a polynomial aerodynamic damping model with structural amplitude as the variable;

[0022] The analytical solution construction module is used to establish an analytical solution of the amplitude probability density distribution function based on the analytical solution of random wind vibration. The analytical solution of the amplitude probability density distribution function is a function that includes the coefficients of structural damping, buffeting force external load, and the polynomial aerodynamic damping model.

[0023] The data acquisition module is used to acquire the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values ​​for the same vibration steady state.

[0024] The fitting and identification module is used to simultaneously fit two sets of amplitude probability density distribution data obtained by the analytical solution of the amplitude probability density distribution function in the same vibration steady state, so as to identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

[0025] The output module is used to output an estimated value of the nonlinear aerodynamic damping for each steady state based on the identified coefficients of the polynomial aerodynamic damping model.

[0026] Furthermore, the fitting and identification module uses the least squares method for fitting to identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

[0027] The present invention also proposes a storage medium on which a computer program is stored, which, when executed by a processor, implements the multistable vortex aerodynamic damping estimation method based on PDF analytical solution as described above.

[0028] The beneficial effects of this invention are as follows:

[0029] This invention relates to a method for estimating aerodynamic damping of multistable vortex-induced vibrations based on PDF analytical solutions. By introducing amplitude probability distribution data under two sets of structural damping conditions for joint inversion, it achieves efficient and accurate identification of strongly nonlinear and non-monotonic aerodynamic damping in multistable vortex-induced vibrations, with significant technical effects. The main components include the following three points.

[0030] (1) It solves the fundamental problem of parameter non-uniqueness in traditional methods for nonlinear identification. The analytical solution derived from the theory of random vibration shows that the amplitude probability distribution under a single damping corresponds to infinitely many combinations of aerodynamic damping and external load. This invention constructs a set of equations containing common aerodynamic damping parameters by adding a set of damping conditions, which guarantees the uniqueness of the coefficients to be identified from a mathematical perspective, thereby breaking through the technical bottleneck of traditional single-data source inversion methods.

[0031] (2) It achieves accurate capture and efficient calculation of complex nonlinear characteristics. This invention directly utilizes the statistical information of amplitude probability distribution and fits it through analytical solutions, avoiding complex time-domain iterations or repetitive experiments. In engineering practice, only two sets of damping conditions need to be tested or simulated under specific steady-state conditions to fully reconstruct the complex non-monotonic relationship of aerodynamic damping with amplitude variation, significantly improving identification efficiency and reliability.

[0032] (3) It provides a direct and effective analytical tool for the wind vibration safety assessment of major engineering structures. This invention can be directly applied to the health monitoring data or wind tunnel test results of actual structures such as bridges and high-rise buildings, accurately identifying the key aerodynamic damping characteristics that lead to multi-steady-state jump phenomena, thus providing crucial parameters for wind-resistant design, vibration control and safety evaluation, and has important theoretical value and engineering application prospects. Attached Figure Description

[0033] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0034] Figure 1 This is a schematic diagram of a wind tunnel test for separating a double box girder from vortex-induced vibration. Figure 1 (a) is a schematic diagram of the model cross-section. Figure 1 (b) is a schematic diagram of the segmental model spring suspension system;

[0035] Figure 2 This is a flowchart of the multistable vortex aerodynamic damping estimation method based on PDF analytical solution of the present invention;

[0036] Figure 3 For the fifth-order polynomial aerodynamic damping model with respect to amplitude;

[0037] Figure 4 The time history of the lift moment coefficient of the structure is used to characterize the buffeting force external load caused by turbulence;

[0038] Figure 5 The random crosswind response data are given under different structural damping conditions; where: (a) represents the structural damping. 5th-order polynomial aerodynamic damping, reduced wind speed (a) Random crosswind response data; (b) Structural damping 5th-order polynomial aerodynamic damping, reduced wind speed (c) Random crosswind response data; 5th-order polynomial aerodynamic damping, reduced wind speed Random crosswind response data;

[0039] Figure 6 For the structure, respectively in structural damping and Comparison of amplitude probability density distribution function data and theoretical values ​​of analytical solutions for random wind-induced vibrations;

[0040] Figure 7 The data is for the amplitude probability density function; (a) shows the structure in a small amplitude stable state, where the analytical solution of the amplitude probability density function is used to simultaneously fit the structural damping. and (a) The case of amplitude probability density distribution function data; (b) The case of fitting the structural damping using the analytical solution of amplitude probability density distribution function under the large amplitude stable state. and The situation regarding the amplitude probability density distribution function data;

[0041] Figure 8 Based on the polynomial coefficients of the identified aerodynamic damping model and load Calculate the aerodynamic damping results and compare them with the target value;

[0042] Figure 9 The fitting results of the amplitude probability density distribution function for aerodynamic damping models of different orders are shown.

[0043] Figure 10 A comparison of aerodynamic damping identification results and target values ​​for aerodynamic damping models of different orders. Detailed Implementation

[0044] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0045] This embodiment takes a separated double box girder as an example. Using the multistable vortex-induced aerodynamic damping estimation method based on PDF analytical solutions of this invention, the nonlinear aerodynamic damping of a bridge tower structure is calculated. The effectiveness and accuracy of the estimated nonlinear aerodynamic damping are evaluated by comparing it with the true value. The bridge tower structure is as follows: Figure 1 As shown.

[0046] Specifically, such as Figure 2As shown in the figure, this embodiment is a multistable vortex vibration aerodynamic damping estimation method based on PDF analytical solution, which includes the following steps.

[0047] Step 1: Based on the multiple stable states in multi-steady vortex-induced vibration, construct a polynomial aerodynamic damping model with structural amplitude as the variable. Specifically, the polynomial aerodynamic damping model is expressed as:

[0048]

[0049] in: Nonlinear aerodynamic damping; air density; The width of the structure; Indicates the effective structural mass per unit length; The amplitude of the structural vibration; These are the polynomial coefficients of the polynomial aerodynamic damping model, and all of them are constants; Let be the order of the polynomial.

[0050] Specifically, this embodiment uses a fifth-order polynomial aerodynamic damping model with respect to amplitude as an example, which is expressed as follows:

[0051]

[0052] The total system damping is the sum of structural damping and aerodynamic damping. When the total damping is zero, the structure reaches a stable vibration state. For example... Figure 3 As shown, the blue curve represents aerodynamic damping. The red dashed line represents structural damping. To visually illustrate structural damping and aerodynamic damping The stable amplitude when the sum is zero will dampen the structure. Draw the red dashed line after taking the negative value. Thus, the structural damping can be obtained through the intersection point between the blue curve and the red dashed line. and aerodynamic damping The stable amplitude when the sum is zero. Specifically, in structural damping... When the total damping is zero, there is only one stable amplitude. In structural damping When the total damping is zero, there are three stable amplitudes in ascending order, including the two largest and smallest stable amplitudes. and and an unstable amplitude of intermediate size Releasing the structure from different initial amplitudes will result in different stable states. That is, when the initial amplitude is less than... At that time, the structure will vibrate and reach its final stable amplitude. When the initial amplitude is greater than At that time, the structure will vibrate and reach its final stable amplitude. .

[0053] Step Two: Based on the analytical solution of stochastic wind-induced vibration, establish the analytical solution of the amplitude probability density distribution function. The analytical solution of the amplitude probability density distribution function is a function that includes the coefficients of structural damping, buffeting force external load, and the polynomial aerodynamic damping model. Specifically, the process of establishing the analytical solution of the amplitude probability density distribution function is as follows:

[0054] The system energy, or Hamiltonian, is defined as the sum of kinetic and potential energy. For quasi-harmonic vibrations:

[0055]

[0056] in: It is the total mechanical energy of the system, which is the sum of kinetic energy and potential energy; It is the natural circular frequency; The amplitude of the structure.

[0057] Based on the Fokker-Planck equations, the theoretical solution of the amplitude probability density function (PDF) under the combined action of buffeting external load and vortex-induced force can be expressed as:

[0058]

[0059]

[0060]

[0061] in: Let be the damping dissipation function of the system; The average wind speed of the incoming flow; This represents the power spectral density value of the lift moment coefficient at the structural frequency; Empirical correction factors representing mode shapes; Dimensionless parameters representing mode shapes.

[0062] Polynomial aerodynamic damping model Substituting these values ​​into the above equation yields the analytical solution for the amplitude probability density distribution function:

[0063]

[0064] in: This represents the amplitude probability density distribution function. PDF is also available express; It is the natural circular frequency; It is a buffeting force external load; It is structural damping; yes The reduction constant, i.e., the normalization constant, is obtained by applying... Regarding the amplitude integral, the integral result is obtained as follows: It equals 1 divided by the integral result. For example... Figure 4 The figure shows the time history data of the external load of the fluttering force.

[0065] When a bridge experiences multi-steady-state vortex-induced vibration, the bridge's response under the combined action of vortex-induced force and buffeting force may transition between small-amplitude and large-amplitude steady states, such as... Figure 5 As shown.

[0066] Figure 5 (a) represents structural damping. 5th-order polynomial aerodynamic damping, reduced wind speed The random crosswind response data shows two steady states, corresponding to stable amplitudes. and Furthermore, the two stable states can transform into each other under the action of external loads.

[0067] Figure 5 (b) is structural damping 5th-order polynomial aerodynamic damping, reduced wind speed The random crosswind response data has only one steady state and one steady amplitude.

[0068] Figure 5 (c) represents structural damping. 5th-order polynomial aerodynamic damping, reduced wind speed The random crosswind response data has two stable states and two stable amplitudes.

[0069] Step 3: For the same vibration steady state, obtain the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values, respectively. Specifically, the first set of structural damping values ​​and the second set of structural damping values ​​are different values.

[0070] Obtaining the amplitude probability density distribution corresponding to different damping levels under the same vibration steady state can be achieved primarily through wind tunnel testing and numerical simulation. In wind tunnel testing, keeping other experimental conditions constant, the damping coefficients of additional structural dampers (such as tuned mass dampers or liquid dampers) are adjusted to set as the first and second group of structural damping values, respectively. Amplitude response data under the corresponding conditions are then measured and extracted, and the probability density distribution is finally obtained through statistical analysis. In numerical simulation, after establishing a finite element model or dynamic model of the structure, the first and second group of structural damping values ​​can be directly set in the simulation parameters. Time history analysis or random vibration analysis is then performed, outputting amplitude time history data, which is then post-processed and statistically analyzed to obtain the desired probability density distribution.

[0071] In this embodiment, structural damping is established in the small amplitude stable state. and Establish structural damping in a large amplitude stable state and The vibration time histories of the bridge under various structural damping conditions and fifth-order aerodynamic damping can be obtained. For example... Figure 6 As shown, the structure is subjected to different conditions depending on the structural damping. and A comparison of amplitude probability density distribution function data and theoretical values ​​of analytical solutions for random wind-induced vibrations. Specifically, structural damping. The experimental values ​​of the amplitude probability density functions for the two corresponding steady states, and the theoretical solutions for the amplitude probability density functions for the two steady states. Additionally, structural damping is considered. A comparison of experimental and theoretical values ​​of amplitude probability density.

[0072] Step 4: In the same vibration steady state, using the analytical solution of the amplitude probability density distribution function established in Step 2, simultaneously fit the two sets of amplitude probability density distribution data obtained in Step 3 to synchronously identify the coefficients of the polynomial aerodynamic damping model and the external buffeting force. In this embodiment, the least squares method is used for fitting to identify the coefficients of the polynomial aerodynamic damping model and the external buffeting force.

[0073] like Figure 7 As shown, in this embodiment, aerodynamic damping models of orders one to five are used. For each vibration steady state, the amplitude probability density distribution data under two structural damping conditions are simultaneously fitted using the analytical solution of stochastic wind vibration, thereby obtaining the aerodynamic damping model coefficients. and load Specifically, Figure 7 (a) is the analytical solution of the amplitude probability density distribution function used to fit the structural damping when the structure is in a stable state with small amplitude. and The situation regarding the amplitude probability density distribution function data. Figure 7 (b) To fit the structural damping using the analytical solution of the amplitude probability density distribution function under the large amplitude stable state. and The amplitude probability density distribution function data is shown below. The least squares method is used for fitting, and the unknowns are the polynomial coefficients of the aerodynamic damping model. and load By simultaneously fitting amplitude probability density data under two structural damping conditions, the uniqueness of the obtained aerodynamic damping curve can be guaranteed.

[0074] Step 5: Based on the coefficients of the identified polynomial aerodynamic damping model, obtain the estimated value of nonlinear aerodynamic damping for each steady state. Place the aerodynamic damping identification results corresponding to the amplitude under different steady states into the same horizontal coordinate system to obtain the identification results for multiple steady states.

[0075] like Figure 8 As shown, these are the polynomial coefficients of the aerodynamic damping model obtained from the identification. and load The calculated aerodynamic damping results were compared with the target value. A fifth-order damping model was used. It was found that the identified aerodynamic damping results have high accuracy. In each steady state, the identified aerodynamic damping curve passes through the two stable amplitudes corresponding to the two structural damping conditions.

[0076] like Figure 9 The figure shows the fitting results of the amplitude probability density distribution function for aerodynamic damping models of different orders. In reality, the order of the aerodynamic damping model cannot be known in advance, therefore... Figure 9 This is used to evaluate the error in the fitted amplitude probability density distribution data caused by the error in the pre-assumed damping model order, and to further study the error in aerodynamic damping identification caused by the assumed damping model order.

[0077] like Figure 10 The figure shows a comparison between the aerodynamic damping identification results and target values ​​for aerodynamic damping models of different orders. It can be observed that in each steady state, the identified aerodynamic damping curves gradually converge as the order of the damping model increases. For small-amplitude steady states, the aerodynamic damping identification results begin to converge when the model order is greater than 3; for large-amplitude steady states, the aerodynamic damping identification results begin to converge when the model order is greater than 4.

[0078] This embodiment also proposes a multistable vortex-induced vibration aerodynamic damping estimation system based on PDF analytical solutions, used to implement the multistable vortex-induced vibration aerodynamic damping estimation method based on PDF analytical solutions described above. Specifically, the multistable vortex-induced vibration aerodynamic damping estimation system based on PDF analytical solutions in this embodiment includes a model building module, an analytical solution building module, a data acquisition module, a fitting and identification module, and an output module. The model building module is used to construct a polynomial aerodynamic damping model with structural amplitude as the variable. The analytical solution building module is used to establish an analytical solution of the amplitude probability density distribution function based on the analytical solution of stochastic wind vibration, wherein the analytical solution of the amplitude probability density distribution function is a function containing structural damping, buffeting force external load, and the coefficients of the polynomial aerodynamic damping model. The data acquisition module is used to acquire the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values ​​for the same vibration steady state. The fitting and identification module is used to simultaneously fit two sets of amplitude probability density distribution data obtained from the analytical solution of the amplitude probability density distribution function in the same vibration steady state, in order to identify the coefficients of the polynomial aerodynamic damping model and the external buffeting force. In this embodiment, the fitting and identification module uses the least squares method for fitting to identify the coefficients of the polynomial aerodynamic damping model and the external buffeting force. The output module is used to output the estimated value of the nonlinear aerodynamic damping in each steady state based on the identified coefficients of the polynomial aerodynamic damping model.

[0079] This embodiment also proposes a storage medium on which a computer program is stored. When the computer program is executed by a processor, it implements the multistable vortex aerodynamic damping estimation method based on PDF analytical solution as described above.

[0080] The embodiments described above are merely preferred embodiments for fully illustrating the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for estimating the aerodynamic damping of multistable vortex vibrations based on PDF analytical solutions, characterized in that: Includes the following steps: Step 1: Based on the multiple stable states in multi-steady vortex-induced vibration, construct a polynomial aerodynamic damping model with structural amplitude as the variable; Step 2: Based on the analytical solution of random wind vibration, establish the analytical solution of the amplitude probability density distribution function. The analytical solution of the amplitude probability density distribution function is a function that includes the coefficients of structural damping, buffeting force external load, and the polynomial aerodynamic damping model. Step 3: For the same vibration steady state, obtain the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values ​​respectively; Step 4: In the same vibration steady state, the analytical solution of the amplitude probability density distribution function established in Step 2 is used to simultaneously fit the two sets of amplitude probability density distribution data obtained in Step 3, so as to simultaneously identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force. Step 5: Based on the coefficients of the identified polynomial aerodynamic damping model, obtain the estimated value of nonlinear aerodynamic damping for each steady state. Place the aerodynamic damping identification results corresponding to the amplitude under different steady states into the same horizontal coordinate system to obtain the identification results under multiple steady states. In step one, the polynomial aerodynamic damping model is expressed as: in: Nonlinear aerodynamic damping; air density; The width of the structure; Indicates the effective structural mass per unit length; The amplitude of the structure; These are the polynomial coefficients of the polynomial aerodynamic damping model, and all of them are constants; The order of the polynomial; In step two, the analytical solution of the amplitude probability density distribution function is expressed as: in: This represents the amplitude probability density distribution function; yes The reduction constants of make such that The integral is 1; It is the natural circular frequency; It is a buffeting force external load; It is structural damping.

2. The method for estimating multistable vortex-induced aerodynamic damping based on PDF analytical solutions according to claim 1, characterized in that: In step three, the damping values ​​of the first group of structures and the damping values ​​of the second group of structures are different values.

3. The method for estimating multistable vortex-induced aerodynamic damping based on PDF analytical solutions according to claim 1, characterized in that: In step four, the least squares method is used for fitting to identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

4. A multistable vortex-induced aerodynamic damping estimation system based on PDF analytical solutions, used to implement the multistable vortex-induced aerodynamic damping estimation method based on PDF analytical solutions as described in any one of claims 1-3, characterized in that: include: The model building module is used to build a polynomial aerodynamic damping model with structural amplitude as the variable; The analytical solution construction module is used to establish an analytical solution of the amplitude probability density distribution function based on the analytical solution of random wind vibration. The analytical solution of the amplitude probability density distribution function is a function that includes the coefficients of structural damping, buffeting force external load, and the polynomial aerodynamic damping model. The data acquisition module is used to acquire the amplitude probability density distribution data of the structure under the set first set of structural damping values ​​and the second set of structural damping values ​​for the same vibration steady state. The fitting and identification module is used to simultaneously fit two sets of amplitude probability density distribution data obtained by the analytical solution of the amplitude probability density distribution function in the same vibration steady state, so as to identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force. The output module is used to output an estimated value of the nonlinear aerodynamic damping for each steady state based on the identified coefficients of the polynomial aerodynamic damping model.

5. The multistable vortex-induced aerodynamic damping estimation system based on PDF analytical solution according to claim 4, characterized in that: The fitting and identification module uses the least squares method to fit the model and identify the coefficients of the polynomial aerodynamic damping model and the external load of the fluttering force.

6. A storage medium, characterized in that: It stores a computer program, which, when executed by a processor, implements the multistable vortex aerodynamic damping estimation method based on PDF analytical solutions as described in any one of claims 1-3.

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