CN115046727A - Bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform - Google Patents

Bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform Download PDF

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CN115046727A
CN115046727A CN202210559599.7A CN202210559599A CN115046727A CN 115046727 A CN115046727 A CN 115046727A CN 202210559599 A CN202210559599 A CN 202210559599A CN 115046727 A CN115046727 A CN 115046727A
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王云飞
姜友荣
周敬
王永景
邓厚雄
齐贵军
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China MCC5 Group Corp Ltd
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Abstract

The invention discloses a bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform, which comprises the steps of establishing a parameter relation between nonlinear pneumatic damping models by utilizing a harmonic balance method; obtaining the time-varying amplitude of the vortex vibration displacement time course of the bridge model in the whole vibration development process from zero to stable amplitude by using Hilbert transform, and fitting to obtain a time-varying amplitude curve to obtain the pneumatic damping of the bridge model; and establishing a motion equation of the vertical vortex-induced vibration of the end face of the bridge, and determining the relationship between the pneumatic damping and the vortex-induced force model parameters according to the motion equation, thereby determining the vortex-induced force. The method introduces the pneumatic damping model in the analysis of the high-rise structure to be used for nonlinear simulation analysis of the bridge vortex-induced force, has a simple form and is convenient to calculate, and meanwhile, the method provides a time-varying amplitude envelope curve which accurately captures the vortex vibration development time course by using Hilbert transform and uses the suggested Gopmertz model for fitting, so that the identification processes of the pneumatic damping and the vortex-induced force are simplified.

Description

Bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform
Technical Field
The invention relates to the technical field of bridge engineering, in particular to a bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform.
Background
Vortex-induced vibration (Vortex-induced vibration) is one of the important problems in large-span bridge wind-induced vibration research, the Vortex-induced vibration phenomena of different degrees are generated in a plurality of bridges such as western and western optical bridges, tiger bridges and the like in China, and the wind speed generated by the Vortex-induced vibration is generally low, so that the fatigue damage and pedestrian discomfort of bridge components can be caused when the Vortex-induced vibration occurs frequently, and the service performance of the bridge is directly influenced. The vortex-induced vibration is a complex resonance phenomenon formed by airflow and a bridge structure at a lower wind speed, and has the characteristics of self-excitation and amplitude-limiting vibration. The vortex-induced force can be understood as the action of pneumatic damping from the perspective of structural dynamics, namely, the vortex-induced force introduces nonlinear pneumatic negative damping for the system, the total damping of the structure is smaller than zero under the action of the pneumatic negative damping, vortex-induced vibration starts to develop, when the pneumatic negative damping is equal to the structural damping, the total damping of the structure is equal to zero, and the vortex-induced vibration reaches a stable amplitude state.
The vortex vibration mechanism is very complex, the problem of vortex vibration cannot be analyzed by a complete analytic method at present, a vortex-induced force mathematical model is generally established in actual engineering through wind tunnel experimental technology and theoretical analysis, vortex-induced force parameters are identified, and the empirical model can accurately reflect the vortex vibration characteristics of the structure. The existing bridge vortex-induced force model mainly adopts a traditional Scanlan nonlinear model, a model parameter identification method is relatively complex, the vortex-induced force model cannot well reverse the response of the model, and the amplitude nonlinearity of real vortex-induced force cannot be accurately expressed.
Disclosure of Invention
The invention mainly aims to provide a bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform, and aims to solve the technical problems that the existing bridge vortex-induced force model mainly adopts a traditional Scanlan nonlinear model, the model parameter identification method is relatively complex, the vortex-induced force model cannot well reverse the response of the model, and the amplitude nonlinearity of real vortex-induced force cannot be accurately expressed.
In order to achieve the above object, the present invention provides a bridge nonlinear vortex-induced force accurate identification method based on hilbert transform, which comprises the following steps:
acquiring a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model, and establishing a parameter relation between the two nonlinear pneumatic damping models by using a harmonic balance method;
obtaining the time-varying amplitude of the vortex vibration displacement time course of the bridge model in the whole vibration development process from zero to stable amplitude by using Hilbert transform, and fitting to obtain a time-varying amplitude curve;
obtaining the pneumatic damping of the bridge model according to the time-varying amplitude curve;
establishing a motion equation of the vertical vortex-induced vibration of the end face of the bridge, and determining the relationship between the pneumatic damping and the vortex-induced force model parameters according to the motion equation;
and determining an expression of the vortex-induced force according to the parameter relationship between the nonlinear aerodynamic damping models and the relationship between the aerodynamic damping and the vortex-induced force model parameters.
Optionally, the time-varying displacement nonlinear pneumatic damping model is ξ a (y)＝B 1 +B 2 |y|+B 3 y 2 The time-varying amplitude nonlinear aerodynamic damping model is
Figure BDA0003653879150000021
Wherein, y and A y Each representing a structureA time-varying displacement and a time-varying amplitude; xi a And xi aeq Respectively representing a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model; b 1 ,B 2 ,B 3 And b 1 ,b 2 ,b 3 Respectively representing the parameters of two nonlinear damping models.
Optionally, a parameter relationship between the two nonlinear aerodynamic damping models is as follows: b is 1 ＝b b ；
Figure BDA0003653879150000022
Figure BDA0003653879150000023
B 3 ＝43 3 。
Optionally, the expression of the time-varying amplitude curve is as follows:
Figure BDA0003653879150000024
wherein A is h Representing a time-varying amplitude, A hm Represents the steady state response amplitude; a is 0 And a 1 Is a constant; t is t c Represents ln (A) h /A hm )＝0.368a 0 The time of day.
Optionally, the expression of the equation of motion of the vertical vortex-induced vibration of the end face of the bridge is as follows:
Figure BDA0003653879150000025
wherein h represents the model vertical displacement; ρ represents an air density; b is B/2, which represents the half width of the bridge section; omega hs Representing the vertical natural vibration circle frequency of the structure; k is ω h b/U represents the reduction frequency; omega h Representing the vertical vibration circle frequency; f VIV Representing a non-linear vortex-induced force;
Figure BDA0003653879150000026
and
Figure BDA0003653879150000027
vortex-induced force model parameters related to aerodynamic damping and aerodynamic stiffness are represented, respectively.
Optionally, the expression of the relationship between the aerodynamic damping and the vortex-induced force model parameters is as follows:
Figure BDA0003653879150000031
wherein ξ ha Is pneumatically damped.
Optionally, the expression of the vortex-induced force is as follows:
Figure BDA0003653879150000032
or
Figure BDA0003653879150000033
The method for accurately identifying the nonlinear vortex-induced force of the bridge based on the Hilbert transform has the following beneficial technical effects:
1. the pneumatic damping model introduced into the high-rise structure analysis is used for nonlinear simulation analysis of bridge vortex-induced force, and is concise in form and convenient to calculate.
2. The method has the advantages that the time-varying amplitude envelope curve of the vortex vibration development time course is accurately captured by using the Hilbert transform, and the fitting is carried out by using the suggested Gopmertz model, so that the identification processes of pneumatic damping and vortex excitation force are simplified.
3. Vortex-induced force parameters to be fitted to the identification
Figure BDA0003653879150000034
Compared with the traditional model, the method points out the defects of the traditional model, and the accuracy of the method is proved through the numerical method for inversely calculating the vortex vibration time course of the bridge model and comparing the vortex vibration time course with the wind tunnel experiment.
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FIG. 1 is a schematic flow chart of a bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform according to the present invention;
FIG. 2 is a schematic view of a wind tunnel model two-degree-of-freedom spring suspension system according to the present invention;
FIG. 3 is a schematic diagram of a vertical vortex vibration displacement experiment time course of the bridge model of the invention;
FIG. 4 is a schematic diagram of a time-varying amplitude envelope fit of the vertical vortex displacement time course of the present invention;
FIG. 5 is a schematic diagram of the vertical vortex vibration aerodynamic damping ratio identification result of the present invention;
FIG. 6 is a schematic diagram of the accuracy comparison of the nonlinear vortex-induced force model of the present invention;
FIG. 7 is a schematic diagram showing comparison of time-course calculation results of vortex vibration displacements according to the present invention.
The implementation, functional features and advantages of the present invention will be further described with reference to the accompanying drawings.
Detailed Description
It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The vortex vibration mechanism is very complex, the problem of vortex vibration cannot be analyzed by a complete analytic method at present, a vortex-induced force mathematical model is generally established in actual engineering through wind tunnel experimental technology and theoretical analysis, vortex-induced force parameters are identified, and the empirical model can accurately reflect the vortex vibration characteristics of the structure. The existing bridge vortex-induced force model mainly adopts a traditional Scanlan nonlinear model, a model parameter identification method is relatively complex, the vortex-induced force model cannot well reverse the response of the model, and the amplitude nonlinearity of real vortex-induced force cannot be accurately expressed.
In order to solve the problem, various embodiments of the bridge nonlinear vortex-induced force accurate identification method based on the Hilbert transform are provided.
The embodiment of the invention provides a bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform, and referring to FIG. 1, FIG. 1 is a flow diagram of the bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform.
In this embodiment, the method for accurately identifying the nonlinear vortex-induced force of the bridge based on the hilbert transform includes the following steps:
s100, acquiring a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model, and establishing a parameter relation between the two nonlinear pneumatic damping models by using a harmonic balance method;
s200, obtaining the time-varying amplitude of the vortex vibration displacement time course of the whole vibration development process of the bridge model from zero to stable amplitude by using Hilbert transform, and fitting to obtain a time-varying amplitude curve;
step S300, obtaining the pneumatic damping of the bridge model according to the time-varying amplitude curve;
s400, establishing a motion equation of the vertical vortex-induced vibration of the end face of the bridge, and determining the relationship between the pneumatic damping and the vortex-induced force model parameters according to the motion equation;
and S500, determining an expression of the vortex-induced force according to the parameter relationship between the nonlinear pneumatic damping models and the relationship between the pneumatic damping and the parameters of the vortex-induced force model.
The method for accurately identifying the bridge nonlinear vortex-induced force based on the hilbert transform is specifically described below by specific examples.
The wind attack angle alpha of a certain large-span bridge is 0deg, the wind speed U is 1.15m/s, and the structural damping ratio xi hs The vertical vortex vibration analysis under 0.32% is taken as an example (the main parameters of the bridge model are that the scale ratio is 1/50, the model width B is 2B is 0.628m, the model height D is 0.06m, the model length L is 2.1m, the mass of the model unit length is 6.524kg/m, and the vertical natural vibration frequency of the model system is f hs 1.294 Hz. ) The contents for explaining the method of the application are as follows:
1. introducing a new bridge nonlinear vortex-induced force model
The application introduces two nonlinear pneumatic damping models adopted in the study of high-rise structure transverse wind vibration in an assessment of evaluation of structural cross wind response with non-ideal lateral vibration damping as an improved bridge nonlinear vortex-induced force model: (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 the time-varying amplitude. Respectively shown in the following two formulas:
ξ a (y)＝B 1 +B 2 |y|+B 3 y 2 (1)
Figure BDA0003653879150000051
wherein, y and A y Time-varying displacements and time-varying amplitudes of structures are denoted herein, respectively; xi a And xi aeq Respectively representing a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model; b is 1 ,B 2 ,B 3 And b 1 ,b 2 ,b 3 Respectively representing the parameters of two nonlinear damping models. Two nonlinear aerodynamic damping models xi can be established based on Harmonic Balance method (Harmonic Balance) a (y) and xi aeq (A y ) The parameter relationship between the two is as follows:
Figure BDA0003653879150000052
it should be noted that the relationship between the two nonlinear aerodynamic damping models established in this embodiment may be established based on a harmonic balancing method, that is, the two aerodynamic damping models are established at a simple harmonic vibration y ═ a y sin (ω t) performs equal work in one vibration cycle, as follows:
Figure BDA0003653879150000053
namely, it is
Figure BDA0003653879150000054
Thereby establishing two nonlinear pneumatic resistancesXi model xi a (y) and xi aeq (A y ) The parameter relationship between the two is as follows:
Figure BDA0003653879150000061
2. method for measuring vortex vibration time-course response of bridge model by utilizing wind tunnel experiment
A free vibration wind tunnel experiment is suspended through a bridge model spring, the whole vibration development time course from zero to stable amplitude of the bridge model is collected, the wind tunnel experiment model is shown in figure 2, and the collected vertical vortex-induced vibration time course response of the bridge is shown in figure 3.
3. Time-varying amplitude of vortex vibration is calculated and fitted by using Hilbert transform
Firstly, a Hilbert Transform (HT) is used to obtain a time-varying amplitude A of a vortex vibration displacement time course h (t), because discrete signals are obtained by directly utilizing HT and need to be fitted into the form of an analytical equation, the invention provides that a Gompertz model is utilized to fit a time-varying amplitude curve of a nonlinear vibration signal, and the model expression is as follows:
Figure BDA0003653879150000062
in the formula, A hm Represents the steady state response amplitude; a is 0 And a 1 Is a constant; t is t c Represents ln (A) h /A hm )＝0.368a 0 The time of day. The time-varying amplitude envelope of the vertical eddy displacement time course obtained by fitting is shown in fig. 4.
It should be noted that, for the hilbert transform calculation, the technical principle is as follows: for a certain nonlinear vibration time-course signal y (t), Hilbert transform is as follows:
Figure BDA0003653879150000063
from which can be obtained
Figure BDA0003653879150000064
Figure BDA0003653879150000065
Figure BDA0003653879150000066
Wherein a (t) and θ (t) are respectively the time-varying amplitude and the time-varying phase of the nonlinear vibration signal y (t), i.e. the instantaneous values corresponding to the respective time steps;
Figure BDA0003653879150000067
4. experimental values for identifying pneumatic damping
And solving and identifying the pneumatic damping by using the time-varying amplitude fitting result of the vortex vibration response. From structural dynamics, it is easy to know that the total damping ratio of the vibration system changing with time after obtaining the time-varying amplitude curve can be solved by the following formula:
Figure BDA0003653879150000071
further, the total damping as a function of amplitude can be found as:
Figure BDA0003653879150000072
the aerodynamic damping can then be obtained by subtracting the structural damping from the total damping, which gives the total damping of the system as a function of time and as a function of amplitude, and the aerodynamic damping, as shown in fig. 5.
5. Parameter identification and fitting precision comparison of vortex-induced force model
The motion equation of the vertical vortex-induced vibration of the bridge section can be expressed as follows:
Figure BDA0003653879150000073
in the formula, h represents the vertical displacement of the model; ρ represents an air density; b is B/2, which represents the half width of the bridge section; omega hs Representing the vertical natural vibration circle frequency of the structure; k is ω h b/U represents the reduction frequency; omega h Representing the vertical vibration circle frequency; f VIV Representing a non-linear vortex-induced force;
Figure BDA0003653879150000074
and
Figure BDA0003653879150000075
vortex-induced force model parameters related to aerodynamic damping and aerodynamic stiffness are represented, respectively.
For single degree of freedom vertical vortex-induced vibration, the aerodynamic stiffness term
Figure BDA0003653879150000076
The influence on the structural frequency is very weak and can be ignored, namely omega h ≈ω hs . As is readily known from equation of motion (7), aerodynamic damping ξ ha And aerodynamic derivative
Figure BDA00036538791500000712
There is a simple relationship between:
Figure BDA0003653879150000077
according to the bridge nonlinear aerodynamic damping model introduced by the invention, the vortex-induced force parameter can be known
Figure BDA0003653879150000078
Can be expressed as a polynomial function of the invariant time-varying amplitude as follows:
Figure BDA0003653879150000079
and is equivalent to a time-varying displacement model expressed as:
Figure BDA00036538791500000710
the expression of the traditional Scanlan vortex-induced force nonlinear empirical model is as follows:
Figure BDA00036538791500000711
wherein epsilon is a constant and can change along with the wind speed; y is 1 (k) Is the aerodynamic derivative.
It is clear that when the Scanlan non-linear vortex force model (11) is expressed as formula (7) suggested herein, there are:
Figure BDA0003653879150000081
comparing the formula (12) and the formula (10), two nonlinear aerodynamic derivative models can be known
Figure BDA0003653879150000082
The following relationships exist:
Figure BDA0003653879150000083
it can be seen that compared with the non-linear model established by introducing the invention, the traditional Scanlan vortex-induced force model lacks a first-order term related to motion, which is a defect of the Scanlan vortex-induced force model.
According to the analysis, the vortex-induced force parameters are obtained by respectively adopting two nonlinear vortex-induced force models of the invention (formula 9) and the traditional Scanlan (formula 12) to identify and fit
Figure BDA0003653879150000084
The results are shown in fig. 6 as a function of amplitude. Therefore, the newly introduced bridge vortex-induced force model is concise in formAnd the amplitude dependence of the vortex-induced force can be captured very accurately, and the precision of the model is greatly improved compared with that of the traditional vortex-induced force model.
6. Vortex-induced vibration time-course response analysis and verification
Using the obtained non-linear vortex-induced force parameters
Figure BDA0003653879150000085
The time-course response of the vortex vibration can be conveniently solved, only the amplitude nonlinear model (formula 9) is required to be converted into the nonlinear model (formula 10) expressed by vibration displacement (or vibration speed), then the displacement time-course response of the two-dimensional vertical vortex-induced vibration is solved by using a numerical method, namely a four-order Runge-Kutta (4th-order Runge-Kutta) method, and compared with the wind tunnel experiment result, the correctness of the analysis method can be verified, and the result is shown in figure 7.
It should be noted that, for the four-step dragon library tower (4th-order Runge-Kutta) method, the technical principle is specifically as follows:
for differential equations:
Figure BDA0003653879150000086
the initial value condition is as follows: y (x) 0 )＝y 0 ；
Then the time course calculation formula adopting the fourth-order Rungestota method is as follows:
Figure BDA0003653879150000087
k 1 ＝Δt·f(x i ,y i ) (22)
Figure BDA0003653879150000088
Figure BDA0003653879150000089
k 4 ＝Δt·f(x i +Δt,y i +k 3 ) (25)
wherein, Δ t is a calculation solving step length; k is a radical of formula 1 ～k 4 The first derivative of the output variable, i.e. the differential at one point, is indicated.
In the embodiment, the method for accurately identifying the bridge nonlinear vortex-induced force based on Hilbert transform is provided, and a pneumatic damping model in high-level structure analysis is introduced for nonlinear simulation analysis of the bridge vortex-induced force, so that the method is simple in form and convenient to calculate; the method has the advantages that a time-varying amplitude envelope curve of the vortex vibration development time course is accurately captured by using Hilbert transform, and fitting is performed by using a suggested Gopmertz model, so that the identification processes of pneumatic damping and vortex excitation force are simplified; finally identifying the fitted vortex-induced force parameters
Figure BDA0003653879150000091
Compared with the traditional model, the method points out the defects of the traditional model, and the accuracy of the method is proved through the numerical method for inversely calculating the vortex vibration time course of the bridge model and comparing the vortex vibration time course with the wind tunnel experiment.
Through the above description of the embodiments, those skilled in the art will clearly understand that the present invention may be implemented by software plus necessary general hardware, and may also be implemented by special hardware including special integrated circuits, special CPUs, special memories, special components and the like. Generally, functions performed by computer programs can be easily implemented by corresponding hardware, and specific hardware structures for implementing the same functions may be various, such as analog circuits, digital circuits, or dedicated circuits. However, the implementation of a software program is a more preferable embodiment for the present invention. Based on such understanding, the technical solutions of the present invention may be embodied in the form of a software product, which is stored in a readable storage medium, such as a floppy disk, a usb disk, a removable hard disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a magnetic disk or an optical disk of a computer, and includes instructions for enabling a computer device (which may be a personal computer, a server, or a network device) to execute the methods according to the embodiments of the present invention.

Claims (7)

1. A bridge nonlinear vortex-induced force accurate identification method based on Hilbert transform is characterized by comprising the following steps of:
acquiring a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model, and establishing a parameter relation between the two nonlinear pneumatic damping models by using a harmonic balance method;
obtaining the time-varying amplitude of the vortex vibration displacement time course of the bridge model in the whole vibration development process from zero to stable amplitude by using Hilbert transform, and fitting to obtain a time-varying amplitude curve;
obtaining the pneumatic damping of the bridge model according to the time-varying amplitude curve;
establishing a motion equation of the vertical vortex-induced vibration of the end face of the bridge, and determining the relationship between the pneumatic damping and the vortex-induced force model parameters according to the motion equation;
and determining the vortex-induced force according to the parameter relationship between the nonlinear pneumatic damping models and the relationship between the pneumatic damping and the vortex-induced force model parameters.
2. The method for accurately identifying the nonlinear vortex-induced force of the bridge based on the Hilbert transform as claimed in claim 1, wherein the time-varying displacement nonlinear aerodynamic damping model is ξ a (y)＝B 1 +B 2 |y|+B 3 y 2 The time-varying amplitude nonlinear aerodynamic damping model is
Figure FDA0003653879140000011
Wherein, y and A y Respectively representing time-varying displacements and time-varying amplitudes of the structure; xi shape a And xi aeq Respectively representing a time-varying displacement nonlinear pneumatic damping model and a time-varying amplitude nonlinear pneumatic damping model; b 1 ,B 2 ,B 3 And b 1 ,b 2 ,b 3 Respectively representing the parameters of two nonlinear damping models.
3. The method for accurately identifying the bridge nonlinear vortex-induced force based on the Hilbert transform as claimed in claim 2, wherein a parameter relationship between the two nonlinear aerodynamic damping models is as follows:
Figure FDA0003653879140000012
4. the method for accurately identifying the nonlinear vortex-induced force of the bridge based on the Hilbert transform according to claim 3, wherein the expression of the time-varying amplitude curve is as follows:
Figure FDA0003653879140000013
wherein A is h Representing a time-varying amplitude, A hm Represents the steady state response amplitude; a is a 0 And a 1 Is a constant; t is t c Represents ln (A) h /A hm )＝0.368a 0 The time of day.
5. The Hilbert transform-based bridge nonlinear vortex-induced force accurate identification method according to claim 4, wherein the expression of the motion equation of the vertical vortex-induced vibration of the end face of the bridge is as follows:
Figure FDA0003653879140000021
wherein h represents the model vertical displacement; ρ represents an air density; b is B/2, which represents the half width of the bridge section; omega hs Representing the vertical natural vibration circle frequency of the structure; k is ω h b/U represents the reduction frequency; omega h Representing the vertical vibration circle frequency; f VIV Representing a non-linear vortex-induced force;
Figure FDA0003653879140000022
and
Figure FDA0003653879140000023
vortex-induced force model parameters related to aerodynamic damping and aerodynamic stiffness are represented, respectively.
6. The Hilbert transform-based bridge nonlinear vortex-induced force accurate identification method according to claim 5, wherein the expression of the relationship between the aerodynamic damping and the vortex-induced force model parameters is as follows:
Figure FDA0003653879140000024
wherein ξ ha Is pneumatically damped.
7. The Hilbert transform-based bridge nonlinear vortex-induced force accurate identification method according to claim 6, wherein the expression of the vortex-induced force is as follows:
Figure FDA0003653879140000025
or
Figure FDA0003653879140000026
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CN115791073A (en) * 2022-10-14 2023-03-14 港珠澳大桥管理局 Pneumatic self-excitation force testing device

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115791073A (en) * 2022-10-14 2023-03-14 港珠澳大桥管理局 Pneumatic self-excitation force testing device
CN115791073B (en) * 2022-10-14 2023-11-28 港珠澳大桥管理局 Pneumatic self-excitation force testing device

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