A precise identification method of nonlinear vortex forces in bridges based on Hilbert transform

Abstract

The present invention discloses a method for accurately identifying nonlinear vortex-induced forces of bridges based on Hilbert transform, the method comprising: using the harmonic balance method to establish a parameter relationship between nonlinear aerodynamic damping models; using Hilbert transform to obtain the time-varying amplitude of the vortex-induced vibration displacement time history of the entire vibration development process of the bridge model from zero to a stable amplitude, and fitting to obtain a time-varying amplitude curve to obtain the aerodynamic damping of the bridge model; establishing a motion equation for the vertical vortex-induced vibration of the bridge end face, and according to the motion equation, determining the relationship between aerodynamic damping and vortex-induced force model parameters, thereby determining the vortex-induced forces. The present invention introduces an aerodynamic damping model in high-rise structure analysis for nonlinear simulation analysis of vortex-induced forces of bridges, with a concise form and convenient calculation. At the same time, it is proposed to use Hilbert transform to accurately capture the time-varying amplitude envelope of the vortex-induced vibration development time history, and use the recommended Gopmertz model for fitting, thereby simplifying the identification process of aerodynamic damping and vortex-induced forces.

Description

Technical Field

[0001] The invention relates to the technical field of bridge engineering, and in particular to a method for accurately identifying nonlinear vortex excitation forces of bridges based on Hilbert transform. Background Art

[0002] Vortex-induced vibration has always been one of the important issues in the study of wind-induced vibration of long-span bridges. Many bridges in China, such as the Xihoumen Bridge and the Humen Bridge, have experienced vortex-induced vibration to varying degrees. Since the wind speed at which vortex-induced vibration occurs is generally low, frequent occurrence will cause fatigue damage to bridge components and discomfort to pedestrians, directly affecting the performance of the bridge. Vortex-induced vibration is a complex resonance phenomenon formed by airflow and bridge structure at low wind speeds. It is a nonlinear wind-induced vibration with self-excited and limited amplitude vibration characteristics. From the perspective of structural dynamics, vortex-induced force can be understood as the effect of aerodynamic damping, that is, vortex-induced force introduces nonlinear aerodynamic negative damping into the system. Under the action of aerodynamic negative damping, the total damping of the structure is less than zero, and vortex-induced vibration begins to develop. When the aerodynamic negative damping is equal to the structural damping, the total structural damping is equal to zero, and the vortex-induced vibration reaches a stable amplitude state.

[0003] The mechanism of vortex vibration is very complex. Currently, it is still impossible to analyze the problem of vortex vibration through a completely analytical method. Generally, in actual engineering, a mathematical model of vortex excitation force is established through wind tunnel experimental technology and theoretical analysis to identify vortex excitation force parameters and enable the empirical model to correctly reflect the vortex vibration characteristics of the structure. The current bridge vortex excitation force model mainly uses the traditional Scanlan nonlinear model, whose model parameter identification method is relatively complex, and the vortex excitation force model cannot well inversely calculate the response of the model and cannot accurately express the amplitude nonlinearity of the real vortex excitation force. Summary of the invention

[0004] The main purpose of the present invention is to provide a method for accurately identifying the nonlinear vortex excitation force of a bridge based on Hilbert transform, aiming to solve the technical problem that the current bridge vortex excitation force model mainly adopts the traditional Scanlan nonlinear model, the model parameter identification method is relatively complex, and the vortex excitation force model cannot well invert the model response and cannot accurately express the amplitude nonlinearity of the real vortex excitation force.

[0005] To achieve the above object, the present invention provides a method for accurately identifying nonlinear vortex-induced forces of bridges based on Hilbert transform, the method comprising the following steps:

[0006] Obtaining a time-varying displacement nonlinear aerodynamic damping model and a time-varying amplitude nonlinear aerodynamic damping model, and establishing a parameter relationship between the two nonlinear aerodynamic damping models using the harmonic balance method;

[0007] By using Hilbert transform, the time-varying amplitude of the vortex-vibration displacement time history of the entire vibration development process of the bridge model from zero to stable amplitude is obtained, and the time-varying amplitude curve is obtained by fitting.

[0008] According to the time-varying amplitude curve, the aerodynamic damping of the bridge model is obtained;

[0009] Establishing the motion equation of the vertical vortex-induced vibration of the bridge end face, and determining the relationship between the aerodynamic damping and the vortex-induced force model parameters according to the motion equation;

[0010] The expression of vortex-induced force is determined according to the parameter relationship between the nonlinear aerodynamic damping models and the relationship between the aerodynamic damping and vortex-induced force model parameters.

[0011] Optionally, the time-varying displacement nonlinear aerodynamic damping model is ξ a (y) = B1 + B2 | y | + B3y 2 , the time-varying amplitude nonlinear aerodynamic damping model is Among them, y and A y denote the time-varying displacement and amplitude of the structure respectively; ξ a and aeq They respectively represent the time-varying displacement nonlinear aerodynamic damping model and the time-varying amplitude nonlinear aerodynamic damping model; B1, B2, B3 and b1, b2, b3 respectively represent the parameters of the two nonlinear damping models.

[0012] Optionally, the parameter relationship between the two nonlinear aerodynamic damping models is: B1 = b b ; B3=433.

[0013] Optionally, the expression of the time-varying amplitude curve is:

[0014]

[0015] Among them, A h represents the time-varying amplitude, A hm represents the steady-state response amplitude; a0 and a1 are constants; t c means ln(A h / A hm )=0.368a0.

[0016] Optionally, the motion equation of the vertical vortex-induced vibration of the bridge end surface is expressed as:

[0017]

[0018] Where h represents the vertical displacement of the model; ρ represents the air density; b = B / 2 represents the half width of the bridge section; ω hsIndicates the vertical natural circular frequency of the structure; k = ω h b / U represents the reduction frequency; ω h Indicates the vertical vibration circular frequency; F VIV represents the nonlinear vortex excitation force; and represent the vortex-induced force model parameters related to aerodynamic damping and aerodynamic stiffness respectively.

[0019] Optionally, the relationship between the aerodynamic damping and the vortex-induced force model parameters is expressed as:

[0020]

[0021] Among them, ξ ha Pneumatic damping.

[0022] Optionally, the expression of the vortex force is:

[0023]

[0024] or

[0025] The embodiment of the present invention proposes a method for accurately identifying nonlinear vortex excitation forces of a bridge based on Hilbert transform, which has the following beneficial technical effects:

[0026] 1. The aerodynamic damping model used in high-rise structure analysis is introduced for nonlinear simulation analysis of bridge vortex excitation, which is simple in form and easy to calculate.

[0027] 2. It is proposed to use Hilbert transform to accurately capture the time-varying amplitude envelope of the vortex-oscillation development history, and use the recommended Gopmertz model for fitting, thereby simplifying the identification process of aerodynamic damping and vortex-induced forces.

[0028] 3. Identify and fit the vortex force parameters Compared with the traditional model, the shortcomings of the traditional model were pointed out. The vortex-vibration time history of the bridge model was back-calculated using numerical methods and compared with wind tunnel experiments, proving the correctness of this method. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 It is a flow chart of a method for accurately identifying nonlinear vortex excitation force of a bridge based on Hilbert transform of the present invention;

[0030] Figure 2 A schematic diagram of a two-degree-of-freedom spring suspension system of a wind tunnel model of the present invention;

[0031] Figure 3 It is a schematic diagram of the vertical vortex-vibration displacement test time history of the bridge model of the present invention;

[0032] Figure 4 It is a schematic diagram of fitting the time-varying amplitude envelope of the vertical vortex vibration displacement time history of the present invention;

[0033] Figure 5 A schematic diagram of the identification result of the vertical vortex aerodynamic damping ratio of the present invention;

[0034] Figure 6 It is a schematic diagram of the accuracy comparison of the nonlinear vortex excitation force model of the present invention;

[0035] Figure 7 It is a schematic diagram for comparing the vortex vibration displacement time history calculation results of the present invention.

[0036] The realization of the purpose, functional features and advantages of the present invention will be further explained in conjunction with embodiments and with reference to the accompanying drawings. DETAILED DESCRIPTION

[0037] It should be understood that the specific embodiments described herein are only used to explain the present invention, and are not used to limit the present invention.

[0038] The mechanism of vortex vibration is very complex. Currently, it is still impossible to analyze the problem of vortex vibration through a completely analytical method. Generally, in actual engineering, a mathematical model of vortex excitation force is established through wind tunnel experimental technology and theoretical analysis to identify vortex excitation force parameters and enable the empirical model to correctly reflect the vortex vibration characteristics of the structure. The current bridge vortex excitation force model mainly uses the traditional Scanlan nonlinear model, whose model parameter identification method is relatively complex, and the vortex excitation force model cannot well inversely calculate the response of the model and cannot accurately express the amplitude nonlinearity of the real vortex excitation force.

[0039] In order to solve this problem, various embodiments of the method for accurately identifying nonlinear vortex excitation force of a bridge based on Hilbert transform of the present invention are proposed.

[0040] The embodiment of the present invention provides a method for accurately identifying nonlinear vortex excitation force of a bridge based on Hilbert transform. Figure 1 , Figure 1 It is a schematic flow chart of the method for accurately identifying nonlinear vortex excitation force of a bridge based on Hilbert transform of the present invention.

[0041] In this embodiment, the Hilbert transform-based accurate identification method of bridge nonlinear vortex excitation force includes the following steps:

[0042] Step S100, obtaining a time-varying displacement nonlinear aerodynamic damping model and a time-varying amplitude nonlinear aerodynamic damping model, and establishing a parameter relationship between the two nonlinear aerodynamic damping models using a harmonic balance method;

[0043] Step S200, using Hilbert transform, obtaining the time-varying amplitude of the vortex vibration displacement time history of the entire vibration development process of the bridge model from zero to a stable amplitude, and fitting to obtain a time-varying amplitude curve;

[0044] Step S300, obtaining the aerodynamic damping of the bridge model according to the time-varying amplitude curve;

[0045] Step S400, establishing a motion equation for vertical vortex-induced vibration of the bridge end face, and determining the relationship between aerodynamic damping and vortex-induced force model parameters based on the motion equation;

[0046] Step S500, determining an expression of vortex-induced force according to the parameter relationship between the nonlinear aerodynamic damping models and the relationship between the aerodynamic damping and vortex-induced force model parameters.

[0047] The following is a specific example of the method for accurately identifying nonlinear vortex-induced forces in bridges based on Hilbert transform in this application.

[0048] Take a long-span bridge at a wind attack angle α = 0 degrees, wind speed U = 1.15 m / s, and structural damping ratio ξ hs = 0.32% as an example (the main parameters of the bridge model are: scale ratio is 1 / 50, model width B = 2b = 0.628m, model height D = 0.06m, model length L = 2.1m; the mass per unit length of the model is m = 6.524kg / m; the vertical natural frequency of the model system is f hs =1.294Hz. ), the content of the present application method is as follows:

[0049] 1. Introducing a new bridge nonlinear vortex force model

[0050] This application introduces two nonlinear aerodynamic damping models used in the paper "Estimation of stochastic crosswind response of wind-excited tall buildings with nonlinear aerodynamic damping" to study the crosswind vibration of high-rise structures as improved bridge nonlinear vortex excitation models: (1) The first model is expressed as a nonlinear function model of time-varying displacement; (2) The second model is expressed as a polynomial function model of time-varying amplitude. They are shown in the following two equations respectively:

[0051] ξ a (y) = B1 + B2 | y | + B3y 2 (1)

[0052]

[0053] Among them, y and A y Here, they represent the time-varying displacement and amplitude of the structure respectively; ξ a andaeq Respectively represent the time-varying displacement nonlinear aerodynamic damping model and the time-varying amplitude nonlinear aerodynamic damping model; B1, B2, B3 and b1, b2, b3 represent the parameters of the two nonlinear damping models. Based on the harmonic balance method (Harmonic Balance), two nonlinear aerodynamic damping models ξ can be established a (y) and ξ aeq (A y ) are as follows:

[0054]

[0055] It should be noted that the relationship between the two nonlinear aerodynamic damping models established in this embodiment can be established based on the harmonic balance method, that is, the two aerodynamic damping models are in the simple harmonic vibration y=A y The work done in one vibration period of sin(ωt) is equal, so:

[0056]

[0057] Right now

[0058]

[0059] Thus, two nonlinear aerodynamic damping models ξ are established. a (y) and ξ aeq (A y ) are as follows:

[0060]

[0061] 2. Using wind tunnel experiments to measure the vortex-induced vibration time-history response of bridge models

[0062] Through the free vibration wind tunnel test of the bridge model spring suspension, the entire vibration development time of the bridge model from zero to stable amplitude is collected. The wind tunnel test model is as follows Figure 2 As shown in the figure, the time history response of the vertical vortex-induced vibration of the bridge is as follows: Figure 3 shown.

[0063] 3. Use Hilbert transform to calculate and fit the time-varying amplitude of vortex vibration

[0064] First, the time-varying amplitude A of the vortex vibration displacement time history is obtained by using the Hilbert Transform (HT) h (t), since the discrete signal obtained by directly using HT needs to be fitted into the form of an analytical equation, the present invention proposes to use the Gompertz model to fit the time-varying amplitude curve of the nonlinear vibration signal, and the model expression is:

[0065]

[0066] In the formula, A hm represents the steady-state response amplitude; a0 and a1 are constants; t c means ln(A h / A hm )=0.368a0. The time-varying amplitude envelope of the vertical vortex vibration displacement time history obtained by fitting is as follows: Figure 4 shown.

[0067] It should be noted that the technical principle of Hilbert transform calculation is as follows: For a nonlinear vibration time-history signal y(t), its Hilbert transform is:

[0068]

[0069] From it we can get

[0070]

[0071]

[0072]

[0073] Among them, A(t) and θ(t) are the time-varying amplitude and time-varying phase of the nonlinear vibration signal y(t), that is, the instantaneous value corresponding to each time step;

[0074] 4. Identification of experimental values ​​of aerodynamic damping

[0075] The aerodynamic damping is solved and identified by using the time-varying amplitude fitting results of the above vortex vibration response. It is easy to know from structural dynamics that after obtaining the time-varying amplitude curve, the total damping ratio of the vibration system that changes with time can be solved by the following formula:

[0076]

[0077] Furthermore, the total damping that varies with amplitude can be obtained as:

[0078]

[0079] Then, the aerodynamic damping can be obtained by subtracting the structural damping from the total damping, as Figure 5 The total system damping and aerodynamic damping are shown as a function of time and amplitude.

[0080] 5. Vortex excitation model parameter identification and fitting accuracy comparison

[0081] The motion equation of the vertical vortex-induced vibration of the bridge section can be expressed as:

[0082]

[0083] Where h represents the vertical displacement of the model; ρ represents the air density; b = B / 2 represents the half width of the bridge section; ω hs Indicates the vertical natural circular frequency of the structure; k = ω h b / U represents the reduction frequency; ω h Indicates the vertical vibration circular frequency; F VIV represents the nonlinear vortex excitation force; and represent the vortex-induced force model parameters related to aerodynamic damping and aerodynamic stiffness respectively.

[0084] For single-degree-of-freedom vertical vortex-induced vibration, the aerodynamic stiffness term The effect on the structural frequency is very weak and can be ignored, that is, ω h ≈ω hs From the equation of motion (7), it is easy to see that the aerodynamic damping ξ ha and aerodynamic derivatives There is a simple relationship between them:

[0085]

[0086] According to the bridge nonlinear aerodynamic damping model introduced in the present invention, the vortex force parameter It can be expressed as a polynomial function with infinite time-varying amplitude as follows:

[0087]

[0088] And it is equivalent to the time-varying displacement model expressed as follows:

[0089]

[0090] The traditional Scanlan vortex excitation nonlinear empirical model expression is:

[0091]

[0092] Among them, ε is a constant that can change with wind speed; Y1(k) is the aerodynamic derivative.

[0093] Obviously, when the Scanlan nonlinear vortex excitation model (11) is expressed as the formula (7) proposed in this paper, we have:

[0094]

[0095] Comparing formula (12) and formula (10), we can see that the two nonlinear aerodynamic derivative models are The following relationship exists:

[0096]

[0097] It can be seen that compared with the nonlinear model introduced and established in the present invention, the traditional Scanlan vortex excitation model lacks the first-order term related to the motion, which is a shortcoming of the Scanlan vortex excitation model itself.

[0098] According to the above analysis, the vortex excitation force parameters are obtained by using the two nonlinear vortex excitation force models of the present invention (Formula 9) and the traditional Scanlan (Formula 12) to identify and fit. As the amplitude changes, the results are as follows Figure 6 It can be seen that the bridge vortex excitation force model newly introduced by the present invention is not only simple in form, but also can capture the amplitude dependence of the vortex excitation force very accurately, and its accuracy is greatly improved compared with the traditional vortex excitation force model.

[0099] 6. Vortex-induced vibration time-history response analysis and verification

[0100] Using the obtained nonlinear vortex force parameters The time-history response of vortex-induced vibration can be easily solved. It only needs to transform the amplitude nonlinear model (Equation 9) into a nonlinear model expressed by vibration displacement (or vibration velocity) (Equation 10), and then use the numerical method - the 4th-order Runge-Kutta method to solve the displacement time-history response of two-dimensional vertical vortex-induced vibration, and compare it with the wind tunnel test results to verify the correctness of this analysis method. The results are shown in Figure 7 shown.

[0101] It should be noted that the technical principle of the 4th-order Runge-Kutta method is as follows:

[0102] For differential equations: There is an initial value condition: y(x0) = y0;

[0103] The time history calculation formula using the fourth-order Runge-Kutta method is:

[0104]

[0105] k1=Δt·f(x i ,y i ) (twenty two)

[0106]

[0107]

[0108] k4=Δt·f(x i +Δt,y i +k3) (25)

[0109] Among them, Δt is the calculation step size; k1~k4 represent the first-order derivative of the output variable, that is, the differential at a point.

[0110] In this embodiment, a method for accurately identifying the nonlinear vortex-induced force of a bridge based on Hilbert transform is proposed. The aerodynamic damping model in high-rise structure analysis is introduced for nonlinear simulation analysis of the vortex-induced force of the bridge, which is concise in form and convenient for calculation. The Hilbert transform is used to accurately capture the time-varying amplitude envelope of the vortex-induced vibration development time course, and the Gopmertz model is used for fitting, thereby simplifying the identification process of aerodynamic damping and vortex-induced force. Finally, the identified and fitted vortex-induced force parameters are used. Compared with the traditional model, the shortcomings of the traditional model were pointed out. The vortex-vibration time history of the bridge model was back-calculated using numerical methods and compared with wind tunnel experiments, proving the correctness of this method.

[0111] Through the description of the above implementation mode, the technicians in the relevant field can clearly understand that the present invention can be implemented by means of software plus necessary general hardware, and of course, it can also be implemented by special hardware including special integrated circuits, special CPUs, special memories, special components, etc. In general, all functions completed by computer programs can be easily implemented by corresponding hardware, and the specific hardware structure used to implement the same function can also be various, such as analog circuits, digital circuits or special circuits. However, for the present invention, software program implementation is a better implementation mode in more cases. Based on such an understanding, the technical solution of the present invention is essentially or the part that contributes to the prior art can be embodied in the form of a software product, which is stored in a readable storage medium, such as a computer floppy disk, a U disk, a mobile hard disk, a read-only memory (ROM, Read-Only Memory), a random access memory (RAM, Random Access Memory), a disk or an optical disk, etc., including a number of instructions to enable a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in each embodiment of the present invention.

Claims

1. A method for accurately identifying nonlinear vortex excitation forces of bridges based on Hilbert transform, characterized in that: The method comprises the following steps: Obtaining a time-varying displacement nonlinear aerodynamic damping model and a time-varying amplitude nonlinear aerodynamic damping model, and establishing a parameter relationship between the two nonlinear aerodynamic damping models using the harmonic balance method; By using Hilbert transform, the time-varying amplitude of the vortex-vibration displacement time history of the entire vibration development process of the bridge model from zero to stable amplitude is obtained, and the time-varying amplitude curve is obtained by fitting. According to the time-varying amplitude curve, the aerodynamic damping of the bridge model is obtained; Establishing the motion equation of the vertical vortex-induced vibration of the bridge end face, and determining the relationship between the aerodynamic damping and the vortex-induced force model parameters according to the motion equation; Determining the vortex excitation force according to the parameter relationship between the nonlinear aerodynamic damping models and the relationship between the aerodynamic damping and vortex excitation force model parameters; The time-varying displacement nonlinear aerodynamic damping model is ξ a (y) = B1 + B2 | y | + B3y 2 , the time-varying amplitude nonlinear aerodynamic damping model is Among them, y and A y denote the time-varying displacement and amplitude of the structure respectively; ξ a and aeq They represent the time-varying displacement nonlinear aerodynamic damping model and the time-varying amplitude nonlinear aerodynamic damping model respectively; B1, B2, B3 and b1, b2, b3 represent the parameters of the two nonlinear damping models respectively; The parameter relationship between the two nonlinear aerodynamic damping models is: B1=b1; B3=4b3; The expression of the time-varying amplitude curve is: Among them, A h represents the time-varying amplitude, A hm represents the steady-state response amplitude; a0 and a1 are constants; t c means ln(A h / A hm )=0.368a0; t represents time; The expression of the motion equation of the vertical vortex-induced vibration of the bridge end surface is: Where h represents the vertical displacement of the model; ρ represents the air density; b = B / 2 represents the half width of the bridge section; ω hs Indicates the vertical natural circular frequency of the structure; k = ω h b / U represents the reduction frequency; ω h Indicates the vertical vibration circular frequency; F VIV represents the nonlinear vortex excitation force; and represent the vortex force model parameters related to aerodynamic damping and aerodynamic stiffness respectively; The relationship between the aerodynamic damping and vortex force model parameters is expressed as: Among them, ξ ha For pneumatic damping; The expression of the vortex force is: or

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