Improved regular perturbation high-order expansion prediction method and system for critical wind speed of long flexible structure bending-torsional coupling flutter

Abstract

This invention discloses an improved canonical perturbation high-order expansion prediction method and system for the critical flutter wind speed of long, flexible structures. The method includes: Step 1: Acquiring aerodynamic and structural parameters based on the Scanlan self-excited force model; Step 2: Constructing the bending-torsional coupling control matrix equation and introducing small parameters for perturbation expansion; Step 3: Using the improved canonical perturbation method, distinguishing between equal and unequal frequency cases, eliminating long-term terms by constructing eigenvalues, and solving for eigenvalues ​​and modal vectors step by step; Step 4: Deriving explicit analytical solutions for modal frequencies, damping ratios, amplitude ratios, and phase differences; Step 5: Determining the modal frequencies and directly solving for the damping ratio through a single iteration to determine the critical flutter wind speed. This invention decomposes complex nonlinear problems into linear superpositions of components, obtaining explicit closed-form solutions for the parameters. The calculation can be completed in a single iteration, significantly simplifying the analysis process and clearly revealing the influence mechanism of various structural and aerodynamic parameters on flutter instability.

Description

Technical Field

[0001] This invention belongs to the field of structural wind-resistant design and aeroelasticity technology, specifically involving a high-order expansion prediction method and system for the improved regular perturbation of critical wind speed for long flexible structures with bending-torsional coupling. It is applicable to the flutter aerodynamic stability analysis of long-span flexible main beam structures (such as long-span bridges, suspension bridges, cable-stayed bridges, etc.) under wind load. Background Technology

[0002] With the rapid development of transportation infrastructure construction, the spanning capacity of long-span bridges and other flexible structures is constantly improving, and structural systems are becoming lighter, more flexible, and less damped, significantly increasing their sensitivity to wind loads. Flutter, a typical aeroelastic instability phenomenon, is a divergent vibration caused by the coupling of airflow self-excitement force and structural vibration. Once it occurs, it often leads to catastrophic structural damage. Therefore, accurately predicting and assessing the flutter critical wind speed is a crucial aspect of the wind-resistant design and safety assurance of long-span bridges.

[0003] Currently, methods for predicting flutter critical wind speeds are mainly divided into two categories: numerical calculation methods and analytical methods. Among them, complex modal eigenvalue analysis is the most widely used method in numerical calculation. It transforms the differential equations of motion into complex modal eigenvalue problems, and can efficiently output accurate numerical solutions for modal frequencies and damping ratios. However, this method is difficult to intuitively and clearly reveal the intrinsic influence mechanism of structural and aerodynamic parameters on flutter excitation, which is not conducive to parameter sensitivity analysis and mechanism exploration in the preliminary design stage.

[0004] In contrast, analytical methods explicitly describe the influence of aerodynamic and structural parameters on flutter stability through mathematical expressions, giving them a natural advantage in revealing physical mechanisms. Traditional closed-form solutions are typically based on the assumption of low damping, deriving approximate expressions for the modal characteristics of the coupled system by assuming different frequencies for the vertical bending and torsional modes. However, in multivariate iterative solutions, the contributions of each parameter are often implicit in the formulas, making them difficult to express explicitly. More importantly, when dealing with strongly coupled conditions where the initial frequencies of the vertical and torsional modes are equal or very close, traditional perturbation methods require reclassification and discussion of the system equations due to problems such as singular eigenvalue matrices and the inability to effectively eliminate long-term terms, thus increasing the complexity and difficulty of application.

[0005] In recent years, the LP perturbation method, as an effective mathematical analytical tool, has been gradually introduced into the field of wind engineering. For example, an existing method for predicting the critical flutter wind speed of long-span bridges based on the LP perturbation method has successfully derived explicit closed-form solutions for modal frequencies and damping ratios through perturbation expansion and successive solution, significantly improving computational efficiency and mechanistic interpretation capabilities. However, this technical solution only considers the conventional case where the vertical and torsional modal frequencies are not equal, neglecting the special but theoretically significant case where the frequencies are equal (or close). In actual bridge structural design, although designers usually deliberately avoid frequency coincidence, due to factors such as construction errors, structural damage, material aging, or changes in ancillary facilities, bridges in certain operational or specific construction phases may still exhibit situations where the vertical bending and torsional modal frequencies are equal or extremely close. In such cases, the traditional LP perturbation method will face difficulties in solving the problem or even fail.

[0006] Furthermore, existing flutter prediction methods for long and flexible structures such as photovoltaic flexible support systems and transmission line galloping mostly adopt energy balance or frequency domain analysis strategies, failing to provide a unified improved canonical perturbation high-order closed-loop solution system that can be applied to different operating conditions with equal and unequal frequencies.

[0007] Therefore, how to overcome the bottleneck of existing perturbation methods under the condition of equal frequency, establish an explicit closed solution system that does not require classification and discussion, can uniformly and efficiently solve the critical wind speed of bending-torsional flutter under different frequency ratio conditions, and clearly reveal the specific role mechanism of each coupling component in flutter instability has become a key technical problem that needs to be solved by those skilled in the art. Summary of the Invention

[0008] In view of this, the purpose of this invention is to provide an improved regular perturbation high-order expansion prediction method and system for critical wind speed of bending-torsional coupling flutter in long and flexible structures. By distinguishing frequency conditions and deriving explicit analytical solutions, it achieves efficient and accurate prediction and reveals the coupling mechanism, thereby solving the problems of existing closed solutions requiring multiple iterations, failing to clearly explain the influence of parameters, and being complex to handle conditions with equal frequencies.

[0009] To achieve the above objectives, the present invention provides the following technical solution: This invention first proposes an improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures, comprising the following steps: Step 1: Based on the Scanlan linear self-excited force model, design a segmental model wind tunnel test, complete the data acquisition of relevant flutter derivatives based on the wind tunnel test, and collect the physical characteristic parameters of the bridge; Step 2: Construct the bridge motion control matrix equation based on modal displacement, and rewrite the bridge motion control matrix equation in the form of a second-order damped linear oscillator, introducing dimensionless small parameters. Perform perturbation expansion, and then apply the mass, damping, and stiffness matrices according to... The power series expansion yields the perturbation differential equations of various orders; Step 3: Using the improved regular perturbation method, solve the perturbation differential equations obtained in Step 2 step by step from low order to high order. By setting the eigenvalues ​​of each order, the coefficients of the long-term terms in the equations are made zero to eliminate the long-term terms. And distinguish between the two cases where the initial structural frequencies of the vertical and torsional modes are the same, and solve for the eigenvalues ​​and eigenvectors of each modal branch respectively. Step 4: Based on the eigenvalues ​​and eigenvectors obtained in Step 3, derive the explicit solutions for the modal frequencies, damping ratios, amplitude ratios, and phase differences of each modal branch under the two cases of different and the same initial structural frequencies. Step 5: Determine whether the vertical and torsional frequencies of the initial structure are the same. If they are not the same, use the formula solution for the case where the initial structure frequencies are different; if they are the same, use the formula solution for the case where the initial structure frequencies are the same. Solve for the modal frequencies through iterative calculation. Based on the convergence of the modal frequencies, directly solve for the damping ratio, amplitude ratio, and phase difference of each mode. Determine the flutter critical wind speed by analyzing the variation law of the modal damping ratio with the incoming wind speed in each direction.

[0010] Furthermore, in step one, the Scanlan linear self-excited force model is expressed as: in: For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency, and , The frequency of motion; and For flutter derivative, ; and These are vertical displacement and vertical velocity, respectively. and These are torsional displacement and torsional velocity, respectively. The physical characteristic parameters include bridge width. Mass per unit length Moment of inertia Vertical structural frequency Structural frequency in the direction of torsion Damping ratio in the vertical direction Damping ratio in the direction of torsion .

[0011] Furthermore, in step two, the method for constructing the bridge motion control matrix equation is as follows: Establish the dynamic displacement expression for the bridge structure at time [time]. Vertical displacement and torsional displacement The expression was then processed using modal decomposition. Based on the Scanlan linear self-excited force model, the generalized modal force is obtained through integration: in: and These represent the generalized modal forces in the vertical and torsional directions of the bridge, respectively. For bridge span; For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; and These represent vertical and torsional displacements, respectively. Transform the bridge dynamics equations into standard motion control matrix equations: in: , and These are the generalized mass matrix, damping matrix, and stiffness matrix, respectively. and The structural mass refers to the structural mass in the vertical and torsional directions, respectively. and These are the damping ratios in the vertical and torsional directions, respectively. and These are the frequencies in the vertical and torsional directions, respectively. air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency; The frequency of motion; and These are the generalized displacement vector and the force vector, respectively. and They are respectively for time The second-order and first-order solutions; and These represent the generalized displacement vectors in the vertical and torsional directions of the bridge, respectively. and These represent the generalized force vectors in the vertical and torsional directions of the bridge, respectively. and For aerodynamic stiffness and damping matrix; For modal integrals, and It is a modal shape function; , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; For the self-integration of the vertical mode, For the self-integration of the torsional mode, and For the cross-integral of the vertical and torsional modes; and For flutter derivative, .

[0012] Furthermore, the method for solving the perturbation differential equations of each order is as follows: Define a new variable: in: For new variables; For eigenvalues; The original differential equations are transformed, and the governing equations are rewritten in the form of a second-order damped linear oscillator: in: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; It is a generalized displacement vector; and They represent right The second and first derivatives; mass matrix Damping matrix and stiffness matrix According to small parameters Power series expansion: in: and Let these represent the zeroth and first order mass matrices, respectively. and Let represent the zeroth and first order damping matrices, respectively; and These represent the zeroth and first order stiffness matrices, respectively. The frequency of motion; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; eigenvalues reconciliation Expand as The power series: in: , , and , , These are the eigenvalues ​​and eigenvectors of each order expanded according to the order of the small parameter; Substituting the above expansion into the standard motion control matrix equation, we can match... By raising the power of each power, we obtain the perturbation differential equations of each order, which are as follows: Zeroth-order perturbation differential equation: First-order perturbation differential equation: Second-order perturbation differential equation: in: and Represents the zeroth-order eigenvector right The first and second derivatives; and Represents a first-order eigenvector right The first and second derivatives; Represents the second-order eigenvector right The second derivative of .

[0013] Furthermore, the method for solving the eigenvectors and eigenvalues ​​of each modal branch is as follows: Solving the zeroth-order perturbation differential equation yields the zeroth-order eigenvectors and eigenvalues: in: It is a constant related to the initial displacement and velocity; The imaginary unit; and New variables for the defined vertical and torsional modes; and These are the vectors corresponding to the vertical and torsional modes in the zero-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the zero-order eigenvalues; Substituting the zeroth-order solution into the first-order perturbation differential equation, the right-hand side of the equation is combined as follows: Where the zeroth-order eigenvector The matrix before is: in: and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues; By constructing suitable eigenvalues ​​to make the coefficients of each order of long-term terms equal to zero, long-term terms are eliminated; and two cases are distinguished for solution.

[0014] Furthermore, the solution process for the case where the initial structural frequencies of the vertical and torsional modes are different is as follows: Construct first-order eigenvalues: To eliminate long-term terms, set the diagonal terms of the matrix before the zeroth-order vector to zero, and then solve for the first-order eigenvectors: in: The coefficients obtained from the homogeneous equation terms are used to find particular solutions corresponding to eigenvalues ​​of different modal branches. , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; The coefficients of the torsional mode relative to the vertical mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; The coefficients of the vertical mode versus the torsional mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; and This is the vector corresponding to the vertical and torsional modes in the first-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues; Substituting the zeroth and first-order solutions into the second-order perturbation differential equation, and eliminating the long-run term by constructing second-order eigenvalues, we obtain the second-order eigenvector, expressed as: in: The dimensionless coefficients obtained by the chain rule when differentiating are... , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; The dimensionless coefficients from the vertical first-order solution to the torsional second-order equation obtained by the chain rule when the equation is differential; The dimensionless coefficients obtained from the torsional first-order solution to the vertical second-order equation by the chain rule when the equation is differential; and These are the eigenvectors corresponding to the vertical and torsional modes in the second-order eigenvectors; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the second-order eigenvalues; Use small parameters The eigenvalues ​​of each order are combined according to the expansion formula to obtain the eigenvector and eigenvalues: in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. and These are the characteristic values ​​for the vertical and torsional directions of the bridge, respectively. The solution process for the case where the initial structural frequencies of the vertical and torsional modes are the same is as follows: Besides the zeroth-order eigenvector The elements on the diagonal of the matrix should be set to 0, and the coefficients of the terms on the right side of the first-order differential equation should all be 0, resulting in the following relation: because and Not all values ​​can be 0; zero-order eigenvector The determinant of the previous matrix is ​​0: The first-order eigenvectors and first-order eigenvalues ​​are obtained by solving the problem. in: These are the first-order eigenvalues ​​for the vertical and torsional modes; and This is the vector corresponding to the vertical and torsional modes in the first-order vector; when When the second-order solution and second-order eigenvalues ​​are neglected, and the small parameter is considered... After merging, the vertical feature vector is obtained. eigenvectors of the direction of twist and eigenvalues : in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. These are the eigenvalues ​​for the vertical and torsional modes.

[0015] Furthermore, in step four, when the structural frequencies of the vertical and torsional modes are different, the relationship between eigenvalues, modal frequencies, and damping is utilized. The explicit analytical solutions for the frequency and damping ratio of each modal branch are derived: Vertical modal frequency and torsional modal frequency for: Vertical modal damping ratio and torsional modal damping ratio for: in: , representing the sum of the modal frequencies in two directions, i.e. and All are the sum of the squares of the modal frequencies in the vertical and torsional directions; , representing the modal frequency difference between two directions; It is the difference between the square of the frequency in the vertical direction and the square of the frequency in the torsional direction; It is the difference between the square of the frequency in the torsional direction and the square of the frequency in the vertical direction; , representing the difference in damping ratio between single degrees of freedom in two directions; The difference in damping ratio between the vertical and torsional directions is the single-degree-of-freedom damping ratio. The difference in damping ratio between the torsional direction and the vertical direction for a single degree of freedom; , where is the similarity coefficient between various modal shapes; The similarity coefficient between the shapes of the vertical and torsional modes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and These represent the modal frequencies in the vertical and torsional directions, respectively. and For flutter derivative, ; , Indicates the vertical direction of the bridge; Indicates the direction of the bridge's twist; and For dimensionless mass and moment of inertia; The solution vector is represented as a linear combination of modal vectors. Ignoring the particular second-order solution, the amplitude ratio and phase difference are calculated: The amplitude ratio of torsional motion to vertical motion and phase difference satisfy: The solution yields: in: and These represent the vertical coordinates and torsional coordinates of the vertical mode, respectively. The amplitude ratio of vertical motion to torsional motion and phase difference satisfy: The solution yields: in: and These represent the vertical and torsional coordinates of the torsional mode, respectively. This refers to the width of the bridge deck. For the self-integration of the vertical mode; This is the cross-integral of the vertical and torsional modes.

[0016] Furthermore, in step four, when the structural frequencies of the vertical and torsional modes are the same, the modal frequencies are obtained using the relationship between eigenvalues, frequencies, and damping. Damping ratio for: in: and These represent the frequency and damping of the vertical and torsional modes, respectively. The frequency of motion; To calculate the imaginary value in a complex number; To calculate the real value of a complex number; This represents the similarity coefficient between the vertical and torsional modal shapes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; and For flutter derivative, ; Amplitude ratio of torsional mode to vertical mode and phase difference satisfy: The solution yields: Amplitude ratio of vertical mode to torsional mode and phase difference satisfy: The solution yields: in: and It is a constant related to the initial displacement and velocity.

[0017] Furthermore, in step five, the method for determining the critical flutter speed is as follows: 51) Determine whether the vertical and torsional frequencies of the initial structure are the same, select the corresponding explicit analytical solution formula, and solve the modal frequencies through iterative calculation to obtain the solution values ​​of the modal frequencies in each direction; 52) After the modal frequencies converge, the damping ratio, amplitude ratio and phase difference of each mode are directly calculated according to the explicit analytical form given in step four. 53) Gradually increase the incoming wind speed and calculate the modal damping ratio of the structure in the vertical and torsional directions respectively. When the wind speed increases to a certain value, if the modal damping in any direction changes from positive to negative, it indicates that the system has changed from damping energy dissipation to continuously absorbing external energy, and the structure will undergo self-excited vibration. The wind speed corresponding to this time is the flutter critical wind speed.

[0018] This invention also proposes a system for implementing the improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter of long flexible structures as described above, comprising: The data acquisition module is used to design and execute segmental model wind tunnel tests based on the Scanlan linear self-excited force model, and to collect the structural physical parameters and aerodynamic characteristic parameters of the bridge. The dynamic modeling module is used to construct the bridge motion control matrix equations based on modal displacements, complete the small parameter perturbation expansion of the equations, and output the perturbation differential equations of each order. The analytical solution module is used to derive explicit analytical solutions for modal frequencies, damping ratios, amplitude ratios, and phase differences by eliminating long-term terms and considering both equal and unequal frequencies, solving eigenvalues ​​and modal vectors step by step. The wind speed prediction module is used to select the corresponding analytical solution set based on the initial frequency relationship, directly solve the damping ratio after iteratively calculating the modal frequency, and determine the flutter critical wind speed based on the variation law of the damping ratio with wind speed.

[0019] The beneficial effects of this invention are as follows: This invention presents an improved regular perturbation high-order expansion prediction method for the critical wind speed of flexural-torsional coupled flutter in long and flexible structures. By introducing the improved regular perturbation high-order expansion method into bridge flutter analysis, a significant technical breakthrough is achieved. The core advantage of this invention lies in the fact that, through perturbation expansion and long-term term elimination in steps two to four, the complex flexural-torsional coupled nonlinear problem is decomposed into a linear superposition of structural, uncoupled, and coupled components. This yields explicit closed-form solutions for parameters such as modal frequencies and damping ratios, completely overcoming the shortcomings of traditional methods that require multiple iterations and have unclear mechanisms. All parameters can be directly solved by a single iteration of the modal frequencies in step five, greatly simplifying the calculation process and shortening the analysis time.

[0020] More importantly, this invention's method, through explicit analytical expressions, clearly and intuitively reveals the specific contribution mechanism of each structural parameter and flutter derivative to flutter instability, providing a powerful theoretical tool for understanding the physical nature of flutter occurrence and identifying key inducing factors. Finally, the critical wind speed prediction based on the change in the sign of the damping ratio provides an efficient and accurate scientific basis for the wind-resistant design and safe operation and maintenance of long and flexible structures. Attached Figure Description

[0021] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 This is a flowchart of the improved regular perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to the present invention. Figure 2 This is a schematic diagram of the aerodynamic force distribution of a bridge section, which intuitively presents the aerodynamic force acting on the bridge section under bending-torsional coupled motion; Figure 3 This is a schematic diagram of a bridge model test. Figure 4 This is a schematic diagram of the cross-sectional structure of a bridge deck. Figure 5 The characteristics of the flutter derivative variation of the cross section of a bridge deck are shown. Among them, (a) and (d) reflect the variation trend of the flutter derivative related to vertical motion with the incoming wind speed; (b) and (c) reflect the variation law of the flutter derivative related to torsional motion with wind speed. Figure 6 The diagram shows a comparison between the results of complex modal eigenvalue analysis and the two-degree-of-freedom (vertical-torsional) closed-loop solution proposed in this invention. Specifically: (a) the diagram compares the variation of modal frequencies of the vertical and torsional modal branches with wind speed under both complex modal eigenvalue analysis and the vertical-torsional coupled closed-loop solution; (b) the diagram compares the variation of modal damping ratios of the vertical and torsional modal branches with wind speed under the two methods; (c) the diagram compares the variation of modal amplitude ratios of the vertical and torsional modal branches with wind speed under the two calculation methods; and (d) the diagram compares the variation of modal phase difference of the vertical and torsional modal branches with wind speed under the two calculation methods. Figure 7 The contribution of each damping component to the total damping is shown; (a) shows the contribution ratio of structural damping, uncoupled damping and coupled damping to the total damping in the vertical modal branch; (b) shows the contribution of structural damping, uncoupled damping and coupled damping to the total damping in the torsional modal branch. Figure 8 Details of the contribution of each coupling term to the coupled damping are provided; where: (a) is the contribution of the torsional coupling term in the vertical modal branch to the coupled damping; and (b) is the contribution of the vertical coupling term in the torsional modal branch to the coupled damping. Detailed Implementation

[0022] 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.

[0023] This invention aims to propose an improved canonical perturbation high-order expansion prediction method and system for the critical wind speed of flutter in long, flexible structures with bending-torsional coupling. Compared with existing technologies, it has significant advantages. It only requires simple iteration of modal frequencies, and the modal damping ratio and eigenvectors can be directly derived from the calculated modal frequencies, simplifying the calculation process, avoiding traditional complex iterations, and shortening the calculation time. Simultaneously, it decomposes the complex coupled nonlinear problem into a linear superposition of structural, uncoupled, and coupled components, clearly revealing the influence of structural and aerodynamic parameters on aerodynamic instability to aid in understanding the flutter mechanism. This invention not only proposes a scientifically sound prediction method but also includes a supporting system encompassing data acquisition, dynamic... The system implementation scheme of modules such as mechanical model, perturbation method solution and wind speed prediction, with each module working together to form a complete and feasible technical solution, is easy to promote and apply. Its prediction results can serve as a scientific basis for bridge wind-resistant design and operation and maintenance, support structural optimization and operation and maintenance strategy formulation, reduce flutter accidents and economic losses, ensure bridge safety, and clearly explain the impact of coupling effect on flutter. The linear combination of total damping intuitively presents the coupling effect, providing a clear idea for the study of coupling mechanism. For bridge flutter analysis with large frequency intervals, the influence of coupling components on modal frequencies can be ignored, and only their effect on damping ratio can be considered, thereby reducing the analysis difficulty and improving the analysis efficiency.

[0024] Specifically, such as Figure 1 As shown in the figure, the improved regular perturbation high-order expansion prediction method for critical wind speed of bending-torsional coupling flutter in long flexible structures in this embodiment includes the following steps.

[0025] Step 1: Based on Scanlan's theory, establish a Scanlan linear self-excited force model. Following the geometric and dynamic similarity criteria, design a segmental model wind tunnel test. Based on the wind tunnel test, complete the data acquisition of relevant flutter derivatives and collect the physical characteristic parameters of the bridge. These will serve as the basic data source for subsequent analysis and calculation, ensuring the accuracy and comprehensiveness of the parameters.

[0026] This embodiment establishes a Scanlan linear self-excited force model and completes relevant experimental data collection. Specifically, the Scanlan linear self-excited force model is expressed as follows: in: For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency, and , The frequency of motion; and The flutter derivative is a parameter related to the aerodynamic shape of the cross-section and needs to be identified through wind tunnel experiments. ; and These are vertical displacement and vertical velocity, respectively. and These are torsional displacement and torsional velocity, respectively.

[0027] like Figure 2 As shown, this embodiment designs a segmental model wind tunnel test based on the linear self-excited force theory. The test process strictly follows geometric and dynamic similarity criteria, fabricating a high-precision main beam cross-sectional model, and establishing an aeroelastic mechanical model with two degrees of freedom (vertical and torsional) through a spring suspension system. The experiment first conducts a windless calibration test to identify the bridge's physical characteristic parameters, including the bridge width. Mass per unit length Moment of inertia Vertical structural frequency Structural frequency in the direction of torsion Damping ratio in the vertical direction Damping ratio in the direction of torsion . Figure 2 middle, ,Right now Half the width of the bridge; Average wind speed; To reverse direction; Vertical direction; The direction is lateral. Subsequently, in the boundary layer wind tunnel, the wind speed was gradually adjusted to cover the target reduced frequency range, exciting the model to generate small-amplitude free decaying vibrations, and the program sequence of displacement and wind speed was collected simultaneously. Based on experimental observation data and combined with frequency domain system identification methods, eight dimensionless flutter derivatives at different reduced frequencies were collected from the coupled vibration response. It is worth noting that the linear self-excited force model in this embodiment uses the full width of the bridge, while when collecting experimental parameters of the self-excited force model with half the bridge width, the flutter derivative needs to be reduced as follows. , , , , , , and .

[0028] Step 2: Construct the bridge motion control matrix equation based on modal displacement, rewrite the bridge motion control matrix equation in the form of a second-order damped linear oscillator, and introduce dimensionless small parameters. Perform perturbation expansion, and then apply the mass, damping, and stiffness matrices according to... The power series expansion yields perturbation differential equations of various orders, including: first, establishing mathematical expressions for the flutter force and displacement of the bridge structure, and processing these expressions using modal decomposition; then, based on the mathematical expressions of the Scanlan linear self-excited force model obtained in step one, obtaining generalized modal forces through integration, and transforming the bridge dynamic equations into standard motion control matrix equations containing generalized mass, generalized damping, and generalized stiffness matrices; subsequently, defining new variables and transforming the original differential equations to rewrite the control equations; and finally, decomposing the motion differential equations according to the order of small parameters based on the perturbation expansion, preparing for the perturbation method solution.

[0029] (1) Construct the bridge motion control matrix equation.

[0030] In this embodiment, the method for constructing the bridge motion control matrix equation is as follows.

[0031] First, establish the dynamic displacement expression for the bridge structure. The bridge structure at time... Relative displacement, such as vertical displacement and torsional displacement The expression was then processed using modal decomposition. Subsequently, based on Scanlan's basic flutter model from the experiment, For vertical self-excited aerodynamic force per unit length, To determine the self-excited aerodynamic force per unit length, a mathematical expression for the self-excited aerodynamic force of bridge flutter is constructed, and then the generalized modal force is obtained through integration. Specifically, based on the Scanlan linear self-excited force model, the generalized modal force is obtained through integration, expressed as: in: and These represent the generalized modal forces in the vertical and torsional directions of the bridge, respectively. For bridge span; For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; and These represent vertical and torsional displacements, respectively.

[0032] The bridge dynamics equations are transformed into standard motion control matrix equations that include a generalized mass matrix, a generalized damping matrix, and a generalized stiffness matrix: in: , and These are the generalized mass matrix, damping matrix, and stiffness matrix, respectively. and The structural mass refers to the structural mass in the vertical and torsional directions, respectively. and These are the damping ratios in the vertical and torsional directions, respectively. and These are the frequencies in the vertical and torsional directions, respectively. air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency; The frequency of motion; and These are the generalized displacement vector and the force vector, respectively. and They are respectively for time The second-order and first-order solutions; and These represent the generalized displacement vectors in the vertical and torsional directions of the bridge, respectively. and These represent the generalized force vectors in the vertical and torsional directions of the bridge, respectively. and For aerodynamic stiffness and damping matrix; For modal integrals, and It is a modal shape function; , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; For the self-integration of the vertical mode, For the self-integration of the torsional mode, and For the cross-integral of the vertical and torsional modes; and For flutter derivative, .

[0033] (2) Solve the perturbation differential equations of each order.

[0034] Before decomposing the equation using the perturbation method, first define entirely new variables: in: For new variables; These are the eigenvalues.

[0035] The original differential equations are transformed, and the governing equations are rewritten in the form of a second-order damped linear oscillator: in: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; It is a generalized displacement vector; and They represent right The second and first derivatives.

[0036] According to small parameters The mass matrix of a second-order damped linear oscillator in the form of a power series expansion. Damping matrix and stiffness matrix , represented as: in: and Let these represent the zeroth and first order mass matrices, respectively. and Let represent the zeroth and first order damping matrices, respectively; and These represent the zeroth and first order stiffness matrices, respectively. The frequency of motion; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; For the self-integration of the vertical mode, For the self-integration of the torsional mode, and This is the cross-integral of the vertical and torsional modes.

[0037] eigenvalues reconciliation Expand as The power series: in: , , and , , These are the eigenvalues ​​and eigenvectors of each order expanded according to the order of the small parameter; Substituting the above expansion into the standard motion control matrix equation, we can match... The powers of and Combine terms of the same order. Assume the equation is valid for all sufficiently small terms. If all are true, then The coefficients of each power should be zero, and the coefficients of each power are the perturbation differential equations of each order. In this embodiment, according to The powers of the terms are combined. and It is a zero matrix, and we ignore it here. And higher-order equations, the resulting differential equations are as follows.

[0038] Zeroth-order perturbation differential equation: First-order perturbation differential equation: Second-order perturbation differential equation: in: and Represents the zeroth-order eigenvector right The first and second derivatives; and Represents a first-order eigenvector right The first and second derivatives; Represents the second-order eigenvector right The second derivative of .

[0039] Step 3: Using an improved canonical perturbation method, solve the perturbation differential equations obtained in Step 2 step by step from low to high order. Eliminate long-term terms by setting eigenvalues ​​for each order to make the coefficients of the long-term terms in the equations zero. Distinguish between cases where the initial structural frequencies of the vertical and torsional modes are the same, and solve for the eigenvalues ​​and eigenvectors of each modal branch separately. Specifically, based on the perturbation differential equations obtained in Step 2, solve them step by step from low to high order, and eliminate long-term terms that cause errors in the traditional perturbation solution by constructing eigenvalues ​​for each order. Distinguish between cases where the initial frequency ratios are the same, classify and discuss the long-term term problems in each case, and gradually solve for the eigenvalues ​​and eigenvectors corresponding to the zeroth, first, and second orders. Substitute the eigenvalues ​​and eigenvectors of each order into the control equation containing the small parameter ε to finally obtain the eigenvectors and eigenvalues ​​of each modal branch.

[0040] (1) Solve for the eigenvectors and eigenvalues ​​of each modal branch.

[0041] In this embodiment, the method for solving the eigenvectors and eigenvalues ​​of each modal branch is as follows.

[0042] Solving the zeroth-order perturbation differential equation yields the zeroth-order eigenvectors and eigenvalues. Specifically, when solving differential equations of each order, the solutions should be performed sequentially from lowest to highest order. For zeroth-order small parameters (… The corresponding perturbation differential equation, solution The zeroth-order eigenvector corresponding to the initial solution belongs to an uncoupled linear term, and the solution and eigenvalues ​​are: in: It is a constant related to the initial displacement and velocity; The imaginary unit; and New variables for the defined vertical and torsional modes; and These are the vectors corresponding to the vertical and torsional modes in the zero-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the zero-order eigenvalues.

[0043] For first-order small parameters ( The corresponding first-order perturbation differential equation is: Substituting the zeroth-order solution into the first-order perturbation differential equation, and Therefore, the right side of the equation can be combined into: Where the zeroth-order eigenvector The matrix before is: in: and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues.

[0044] In traditional perturbation methods, terms on the diagonal of the matrix tend to increase over time. This term, also called the long-term term, means that for the results obtained from perturbation series expansion, the error gradually increases with time, eventually leading to the failure of the perturbation method.

[0045] In this embodiment, by constructing suitable eigenvalues ​​to make the coefficients of each long-term term equal to zero, i.e., eliminating the long-term terms, the corresponding first-order solution can be obtained. It is worth noting that at this point, it is necessary to distinguish whether the initial structure frequencies in the two directions are equal; the derivation process will begin to show differences after the first order.

[0046] (2) Solution process when the initial structural frequencies of the vertical and torsional modes are different.

[0047] Specifically, when the initial structural frequencies of the vertical and torsional modes are not the same, first-order eigenvalues ​​are constructed: To eliminate long-term terms, set the diagonal terms of the matrix before the zeroth-order vector to zero, and then solve for the first-order eigenvectors: in: The coefficients obtained from the homogeneous equation terms are used to find particular solutions corresponding to eigenvalues ​​of different modal branches. , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; The coefficients of the torsional mode relative to the vertical mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; The coefficients of the vertical mode versus the torsional mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; and This is the vector corresponding to the vertical and torsional modes in the first-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues.

[0048] For second-order small parameters ( The corresponding differential equation is: Substituting the zeroth and first-order solutions into the second-order perturbation differential equation, and eliminating the long-run term by constructing second-order eigenvalues, we obtain the second-order eigenvector, expressed as: in: The dimensionless coefficients obtained by the chain rule when differentiating are... , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; The dimensionless coefficients from the vertical first-order solution to the torsional second-order equation obtained by the chain rule when the equation is differential; The dimensionless coefficients obtained from the torsional first-order solution to the vertical second-order equation by the chain rule when the equation is differential; and These are the eigenvectors corresponding to the vertical and torsional modes in the second-order eigenvectors; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the second-order eigenvalues.

[0049] Use small parameters The eigenvalues ​​of each order are combined according to the expansion formula to obtain the eigenvector and eigenvalues: in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. and These are the characteristic values ​​for the vertical and torsional directions of the bridge, respectively. (3) The solution process when the initial structural frequencies of the vertical and torsional modes are the same.

[0050] In this embodiment, when the initial structural frequencies of the vertical and torsional modes are the same, the form of the solution will differ from that of the first-order differential equation. At this point, besides the zeroth-order eigenvector... The elements on the diagonal of the matrix should be set to 0, and the coefficients of all terms on the right-hand side of the first-order differential equation should be 0. Therefore, the following relation is obtained: Using algebraic knowledge, because and All eigenvectors cannot be 0; the rank of the matrix should be less than 2; zero-order eigenvectors. The determinant of the previous matrix is ​​0: The first-order eigenvectors and first-order eigenvalues ​​are obtained by solving the problem. in: These are the first-order eigenvalues ​​for the vertical and torsional modes. It is worth noting that the formula can only provide values ​​for two directions, but cannot determine the corresponding modal directions. and It is the vector corresponding to the vertical and torsional modes in the first-order vector.

[0051] when When the second-order solution and second-order eigenvalues ​​are neglected, therefore, with the same initial structure frequency at wind speed of 0, the solution only needs to consider the zeroth-order and first-order solutions and eigenvalues. (Based on small parameters) After merging, the vertical feature vector is obtained. eigenvectors of the direction of twist and eigenvalues : in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. These are the eigenvalues ​​for the vertical and torsional modes.

[0052] Step 4: Based on the eigenvalues ​​and eigenvectors obtained in Step 3, derive the explicit solutions for the modal frequencies, damping ratios, amplitude ratios, and phase differences of each modal branch under the two cases of different and the same initial structural frequencies.

[0053] (1) The structural frequencies of the vertical and torsional modes are different.

[0054] When the structural frequencies of the vertical and torsional modes are different, the relationship between eigenvalues, modal frequencies, and damping can be utilized. To make the solution explicit, wind speed and damping can be considered small near the critical condition. Explicit analytical solutions for the frequencies and damping ratios of the modal branches in each direction can then be obtained. For the same initial structural frequencies, the results for frequencies and damping ratios are as follows.

[0055] Vertical modal frequency and torsional modal frequency for: Vertical modal damping ratio and torsional modal damping ratio for: in: , representing the sum of the modal frequencies in two directions; and All are the sum of the squares of the modal frequencies in the vertical and torsional directions; , representing the modal frequency difference between two directions; It is the difference between the square of the frequency in the vertical direction and the square of the frequency in the torsional direction; It is the difference between the square of the frequency in the torsional direction and the square of the frequency in the vertical direction; , representing the difference in damping ratio between single degrees of freedom in two directions; The difference in damping ratio between the vertical and torsional directions is the single-degree-of-freedom damping ratio. The difference in damping ratio between the torsional direction and the vertical direction for a single degree of freedom; , where is the similarity coefficient between various modal shapes; The similarity coefficient between the shapes of the vertical and torsional modes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and These represent the modal frequencies in the vertical and torsional directions, respectively. and For flutter derivative, ; , Indicates the vertical direction of the bridge; Indicates the direction of the bridge's twist; and Let be the dimensionless mass and the moment of inertia.

[0056] When solving for the frequency of a modal branch, the frequency can be simplified. Specifically, the frequency of the coupled modal branch can be approximated as the frequency affected by the uncoupled self-excited force; for example, when solving for the frequency of the vertical modal branch. The remaining modal frequencies in the vertical frequency formula are expressed as follows: .

[0057] solution vector Represented as mode vector ( The linear combination of , neglecting the particular second-order solution under critical conditions, is calculated as follows: in: and These represent the vertical coordinates and torsional coordinates of the vertical mode, respectively. and These represent the vertical coordinates and torsional coordinates of the torsional mode, respectively.

[0058] The amplitude ratio of torsional motion to vertical motion and phase difference satisfy: The solution yields: The amplitude ratio of vertical motion to torsional motion and phase difference satisfy: The solution yields: in: This refers to the width of the bridge deck. For the self-integration of the vertical mode; This is the cross-integral of the vertical and torsional modes.

[0059] (2) The structural frequencies of the vertical and torsional modes are the same. When the structural frequencies of the vertical and torsional modes are the same, the modal frequencies can be obtained using the relationship between eigenvalues, frequencies, and damping. Damping ratio for: in: and These are the frequencies and damping of the vertical and torsional modes, respectively. It is worth noting that the formulas for the modal frequencies and damping obtained from the two methods cannot determine the corresponding directions; they can only provide values ​​for the two directions. The frequency of motion; To calculate the imaginary value in a complex number; To calculate the real value of a complex number; This represents the similarity coefficient between the vertical and torsional modal shapes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; and For flutter derivative, .

[0060] Amplitude ratio of torsional mode to vertical mode and phase difference satisfy: The solution yields: Amplitude ratio of vertical mode to torsional mode and phase difference satisfy: The solution yields: in: and It is a constant related to the initial displacement and velocity.

[0061] Step 5: Determine if the vertical and torsional frequencies of the initial structure are the same. If they are different, use the formula for the case where the initial structure frequencies are different; if they are the same, use the formula for the case where the initial structure frequencies are the same. Both formulas use a single-iteration calculation method to solve for the modal frequencies. Based on the convergence of the modal frequencies, the damping ratio, amplitude ratio, and phase difference of each mode are directly obtained. By analyzing the variation of the modal damping ratio with the incoming wind speed in each direction, the critical wind speed value for bridge flutter is determined.

[0062] The method for determining the critical flutter speed is as follows: 51) Determine whether the vertical and torsional frequencies of the initial structure are the same, and select the corresponding explicit analytical solution formula. Using the modal branches established in step three, solve for the modal frequencies through iterative calculation to obtain the solution values ​​for the modal frequencies in each direction; 52) After the modal frequencies converge, the damping ratio, amplitude ratio, and phase difference of each mode are directly calculated according to the explicit analytical form given in step four. Specifically, after determining the modal frequencies of each order, the expressions for the damping ratio, amplitude ratio, and phase difference can be directly solved according to the explicit analytical form given in step four without further iteration, thus obtaining the complete modal characteristic parameters.

[0063] 53) Finally, determine the critical flutter wind speed and analyze the main inducing factors of flutter instability. Gradually increase the incoming wind speed and calculate the modal damping ratio of the structure in the vertical and torsional degrees of freedom. When the wind speed increases to a certain value, if the modal damping in any direction changes from positive to negative, it indicates that the system has changed from damping energy dissipation to continuously absorbing external energy, and the structure will undergo self-excited vibration. The corresponding wind speed at this time is the critical flutter wind speed.

[0064] This embodiment also proposes a system for implementing the improved canonical perturbation high-order expansion prediction method for the critical flutter wind speed of long flexible structures as described above. The system includes a data acquisition module, a dynamic modeling module, an analytical solution module, and a wind speed prediction module. In this embodiment, the data acquisition module is used to design and execute segmental model wind tunnel tests based on the Scanlan linear self-excited force model, collecting the structural physical parameters and aerodynamic characteristic parameters of the bridge. The dynamic modeling module is used to construct the bridge motion control matrix equation based on modal displacements, complete the small-parameter perturbation expansion of the equations, and output the perturbation differential equations of each order. The analytical solution module uses the improved canonical perturbation method to eliminate long-term terms and solve for eigenvalues ​​and modal vectors step by step, deriving explicit analytical solutions for modal frequencies, damping ratios, amplitude ratios, and phase differences. The wind speed prediction module selects the corresponding analytical solution set based on the initial frequency relationship, directly solves the damping ratio after iteratively calculating the modal frequencies, and determines the flutter critical wind speed based on the variation of the damping ratio with wind speed.

[0065] The specific implementation methods and technical effects of this embodiment will be further explained below with reference to specific examples.

[0066] This embodiment uses a flutter model experiment of a large bridge as the research object. As shown in Figure 4, the bridge is a cable-stayed bridge with a main span of 1088 m, and the main girder adopts a steel box girder section. Figure 5 shows the flutter derivatives identified based on the segmental model wind tunnel test. The segmental model width B = 0.812 m, and the mass per unit length... Moment of inertia per unit length The vertical and torsional frequencies and the damping ratio of the structure are respectively: Hz, Hz, and Among them, Figure 5(a) and Figure 5Figure 5(d) shows the variation of the flutter derivative related to vertical motion with wind speed; Figures 5(b) and (c) show the variation of the flutter derivative related to torsional motion with wind speed. Specifically, Figure 5 In (a), the black curve represents the flutter derivative. The variation with wind speed; the blue curve represents the flutter derivative. The variation pattern with wind speed; Figure 5 In (b), the black curve represents the flutter derivative. The variation with wind speed; the blue curve represents the flutter derivative. The variation pattern with wind speed; Figure 5 In (c), the black curve represents the flutter derivative. The variation with wind speed; the blue curve represents the flutter derivative. The variation pattern with wind speed; Figure 5 In (d), the black curve represents the flutter derivative. The variation with wind speed; the blue curve represents the flutter derivative. The pattern of variation with wind speed.

[0067] Figure 6 compares the modal frequencies, modal damping ratios, amplitude ratios, and phase differences obtained by the traditional eigenvalue analysis method and the closed-loop analytical solution method proposed in this paper. Specifically, Figure 6(a) compares the modal frequencies of each vertical and torsional modal branch with wind speed; Figure 6(b) compares the modal damping ratios of each modal branch with wind speed; Figure 6(c) compares the amplitude ratios between each mode, including the vertical-torsional and torsional-vertical amplitude ratios; and Figure 6(d) compares the phase differences between each mode, including the vertical-torsional and torsional-vertical phase differences. The calculation results show that the results from the two methods are highly consistent within the wind speed range of 0–12 m / s.

[0068] It should be noted that the closed analytical solution in this embodiment is based on the assumption of small damping, thus exhibiting high accuracy near the critical wind speed; however, as the absolute value of the damping ratio further increases, the analytical solution and the eigenvalue numerical solution will gradually deviate. The torsional mode frequency and damping ratio fluctuate with increasing wind speed, while the vertical mode frequency and damping ratio show an increasing trend. When the wind speed reaches 10.6 m / s, the damping ratio of the torsional mode branch changes from positive to negative, indicating that the structure has entered a flutter instability state.

[0069] According to the analytical expression of modal damping ratio, the total aerodynamic damping can be decomposed into three parts: structural damping, uncoupled damping, and coupled damping. Figure 7 shows the contribution ratio of each damping component in the vertical and torsional mode branches under different wind speeds: Figure 7(a) shows the distribution law of structural damping, uncoupled damping, and coupled damping in the vertical mode; Figure 7(b) shows the contribution law of each damping component in the torsional mode. In the calculation, the structural damping is basically a constant value.

[0070] For the vertical mode, the increase in total damping is mainly due to uncoupled aerodynamic damping. Contributions: Within the analyzed wind speed range, uncoupled damping remains positive throughout. Coupled damping from the torsional mode is also positive, and gradually increases with increasing wind speed. For the torsional mode, uncoupled damping... At low wind speeds, coupled damping dominates, but at high wind speeds, it gradually becomes the controlling factor. The trend of uncoupled damping is consistent with the change of the corresponding flutter derivative; coupled damping is negative, and its absolute value increases continuously with increasing wind speed. At the flutter critical state, although both structural damping and uncoupled damping are positive, negative coupled damping dominates, ultimately leading to continuous energy input into the system and aerodynamic instability.

[0071] The closed-loop analytical solution established in this embodiment clearly reveals the intrinsic mechanism of coupled damping because it expresses the total damping as a linear superposition of the physical components. From step four... and As can be seen from the solution formula, the coupled damping includes contributions from another mode, and each type of contribution can be further subdivided into four terms (denoted as terms 1 to 4). Figure 8 shows the percentage contribution of each sub-term to the coupled damping in the vertical and torsional modes at different wind speeds, which can intuitively reflect the coupling mechanism. Among them, Figure 8(a) shows the contribution of the torsional coupling component in the vertical mode. Figure 8 (b) Contribution to the vertical coupling component in the torsional mode.

[0072] In the vertical mode, the coupling effect is mainly dominated by the torsional coupling component, and in step four... and The first term in the solution formula ( ) is the decisive factor. In the torsional mode, the coupling effect is mainly controlled by the vertical coupling component, of which term 2 ( The negative coupling damping that appears in the torsional mode is essentially caused by step four. and The solution formula Key item control.

[0073] For a given natural frequency, the modal coupling effect exhibits opposite effects in different modal branches: for example, vertical-bending-torsional coupling damping exhibits positive damping (stability effect) in the vertical mode, but negative damping (instability effect) in the torsional mode, and the determining factor is also the same. The terms corresponding to the parameters. This type of coupling effect is closely related to the frequency of the target mode itself, which is also an important reason why the coupling effect of torsional modes is more significant.

[0074] The coupling strength is also controlled by the modal frequency spacing: the closer the modal frequencies are, the stronger the coupling damping effect; the greater the frequency difference, the weaker the coupling effect. It is worth noting that when a specific frequency relationship is satisfied, the coupling damping, which was originally negative in the torsional direction, can be converted into positive damping, and the structural stability is significantly improved.

[0075] The above-described embodiments are merely preferred embodiments provided to fully illustrate 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 high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures, characterized in that: Includes the following steps: Step 1: Based on the Scanlan linear self-excited force model, design a segmental model wind tunnel test, complete the data acquisition of relevant flutter derivatives based on the wind tunnel test, and collect the physical characteristic parameters of the bridge; Step 2: Construct the bridge motion control matrix equation based on modal displacement, rewrite the bridge motion control matrix equation in the form of a second-order damped linear oscillator, and introduce dimensionless small parameters. Perform perturbation expansion, and then apply the mass, damping, and stiffness matrices according to... The power series expansion yields the perturbation differential equations of various orders; Step 3: Using the improved regular perturbation method, solve the perturbation differential equations obtained in Step 2 step by step from low order to high order. By setting the eigenvalues ​​of each order, the coefficients of the long-term terms in the equations are made zero to eliminate the long-term terms. And distinguish between the two cases where the initial structural frequencies of the vertical and torsional modes are the same, and solve for the eigenvalues ​​and eigenvectors of each modal branch respectively. Step 4: Based on the eigenvalues ​​and eigenvectors obtained in Step 3, derive the explicit solutions for the modal frequencies, damping ratios, amplitude ratios, and phase differences of each modal branch under the two cases of different and the same initial structural frequencies. Step 5: Determine whether the vertical and torsional frequencies of the initial structure are the same. If they are not the same, use the formula solution for the case where the initial structure frequencies are different; if they are the same, use the formula solution for the case where the initial structure frequencies are the same. Solve for the modal frequencies through iterative calculation. Based on the convergence of the modal frequencies, directly solve for the damping ratio, amplitude ratio, and phase difference of each mode. Determine the flutter critical wind speed by analyzing the variation law of the modal damping ratio with the incoming wind speed in each direction.

2. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 1, characterized in that: In step one, the Scanlan linear self-excited force model is expressed as: in: For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency, and , The frequency of motion; and For flutter derivative, ; and These are vertical displacement and vertical velocity, respectively. and These are torsional displacement and torsional velocity, respectively. The physical characteristic parameters include bridge width. Mass per unit length Moment of inertia Vertical structural frequency Structural frequency in the direction of torsion Damping ratio in the vertical direction Damping ratio in the direction of torsion .

3. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 1, characterized in that: In step two, the method for constructing the bridge motion control matrix equations is as follows: Establish the dynamic displacement expression for the bridge structure at time [time]. Vertical displacement and torsional displacement The expression was then processed using modal decomposition. Based on the Scanlan linear self-excited force model, the generalized modal force is obtained through integration: in: and These represent the generalized modal forces in the vertical and torsional directions of the bridge, respectively. For bridge span; For vertical self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; and These represent vertical and torsional displacements, respectively. Transform the bridge dynamics equations into standard motion control matrix equations: in: , and These are the generalized mass matrix, damping matrix, and stiffness matrix, respectively. and The structural mass refers to the structural mass in the vertical and torsional directions, respectively. and These are the damping ratios in the vertical and torsional directions, respectively. and These are the frequencies in the vertical and torsional directions, respectively. air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency; The frequency of motion; and These are the generalized displacement vector and the force vector, respectively. and They are respectively for time The second-order and first-order solutions; and These represent the generalized displacement vectors in the vertical and torsional directions of the bridge, respectively. and These represent the generalized force vectors in the vertical and torsional directions of the bridge, respectively. and For aerodynamic stiffness and damping matrix; For modal integrals, and It is a modal shape function; , Indicates the vertical direction of the bridge. Indicates the direction of the bridge's twist; For the self-integration of the vertical mode, For the self-integration of the torsional mode, and For the cross-integral of the vertical and torsional modes; and For flutter derivative, .

4. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 3, characterized in that: The method for solving perturbation differential equations of all orders is as follows: Define a new variable: in: For new variables; For eigenvalues; The original differential equations are transformed, and the governing equations are rewritten in the form of a second-order damped linear oscillator: in: This is the quality matrix; Here is the damping matrix; Here is the stiffness matrix; It is a generalized displacement vector; and They represent right The second and first derivatives; mass matrix Damping matrix and stiffness matrix According to small parameters Power series expansion: in: and Let these represent the zeroth and first order mass matrices, respectively. and Let represent the zeroth and first order damping matrices, respectively; and These represent the zeroth and first order stiffness matrices, respectively. The frequency of motion; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; eigenvalues reconciliation Expand as The power series: in: , , and , , These are the eigenvalues ​​and eigenvectors of each order expanded according to the order of the small parameter; Substituting the above expansion into the standard motion control matrix equation, we can match... By raising the power of each power, we obtain the perturbation differential equations of each order, which are as follows: Zeroth-order perturbation differential equation: First-order perturbation differential equation: Second-order perturbation differential equation: in: and Represents the zeroth-order eigenvector right The first and second derivatives; and Represents a first-order eigenvector right The first and second derivatives; Represents the second-order eigenvector right The second derivative of .

5. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 4, characterized in that: The method for solving the eigenvectors and eigenvalues ​​of each modal branch is as follows: Solving the zeroth-order perturbation differential equation yields the zeroth-order eigenvectors and eigenvalues: in: and All of these are constants related to the initial displacement and velocity; The imaginary unit; and New variables for the defined vertical and torsional modes; and These are the vectors corresponding to the vertical and torsional modes in the zero-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the zero-order eigenvalues; Substituting the zeroth-order solution into the first-order perturbation differential equation, the right-hand side of the equation is combined as follows: Where the zeroth-order eigenvector The matrix before is: in: and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues; By constructing suitable eigenvalues ​​to make the coefficients of each order of long-term terms equal to zero, long-term terms are eliminated; and two cases are distinguished for solution.

6. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 5, characterized in that: The solution process for cases where the initial structural frequencies of the vertical and torsional modes are different is as follows: Construct first-order eigenvalues: To eliminate long-term terms, set the diagonal terms of the matrix before the zeroth-order vector to zero, and then solve for the first-order eigenvectors: in: The coefficients of the torsional mode relative to the vertical mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; The coefficients of the vertical mode versus the torsional mode obtained from the homogeneous equation terms when seeking particular solutions corresponding to eigenvalues ​​of different modal branches; and This is the vector corresponding to the vertical and torsional modes in the first-order vector; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the first-order eigenvalues; Substituting the zeroth and first-order solutions into the second-order perturbation differential equation, and eliminating the long-run term by constructing second-order eigenvalues, we obtain the second-order eigenvector, expressed as: in: The dimensionless coefficients from the vertical first-order solution to the torsional second-order equation obtained by the chain rule when the equation is differential; The dimensionless coefficients obtained from the torsional first-order solution to the vertical second-order equation by the chain rule when the equation is differential; and These are the eigenvectors corresponding to the vertical and torsional modes in the second-order eigenvectors; and These are the eigenvalues ​​corresponding to the vertical and torsional modes in the second-order eigenvalues; Use small parameters The eigenvalues ​​of each order are combined according to the expansion formula to obtain the eigenvector and eigenvalues: in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. and These are the characteristic values ​​for the vertical and torsional directions of the bridge, respectively. The solution process for the case where the initial structural frequencies of the vertical and torsional modes are the same is as follows: Besides the zeroth-order eigenvector The elements on the diagonal of the matrix should be set to 0, and the coefficients of the terms on the right side of the first-order differential equation should all be 0, resulting in the following relation: because and Not all values ​​can be 0; zero-order eigenvector The determinant of the previous matrix is ​​0: The first-order eigenvectors and first-order eigenvalues ​​are obtained by solving the problem. in: These are the first-order eigenvalues ​​for the vertical and torsional modes; and This is the vector corresponding to the vertical and torsional modes in the first-order vector; when When the second-order solution and second-order eigenvalues ​​are neglected, and the small parameter is considered... After merging, the vertical feature vector is obtained. eigenvectors of the direction of twist and eigenvalues : in: and These are the feature vectors for the vertical and torsional directions of the bridge, respectively. These are the eigenvalues ​​for the vertical and torsional modes.

7. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 1, characterized in that: In step four, when the structural frequencies of the vertical and torsional modes are different, the relationship between eigenvalues, modal frequencies, and damping is utilized. The explicit analytical solutions for the frequency and damping ratio of each modal branch are derived: Vertical modal frequency and torsional modal frequency for: Vertical modal damping ratio and torsional modal damping ratio for: in: and All are the sum of the squares of the modal frequencies in the vertical and torsional directions; It is the difference between the square of the frequency in the vertical direction and the square of the frequency in the torsional direction; It is the difference between the square of the frequency in the torsional direction and the square of the frequency in the vertical direction; The difference in damping ratio between the vertical and torsional directions is the single-degree-of-freedom damping ratio. The difference in damping ratio between the torsional direction and the vertical direction for a single degree of freedom; The similarity coefficient between the shapes of the vertical and torsional modes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and These represent the modal frequencies in the vertical and torsional directions, respectively. and For flutter derivative, ; and For dimensionless mass and moment of inertia; The solution vector is represented as a linear combination of modal vectors. Ignoring the particular second-order solution, the amplitude ratio and phase difference are calculated: The amplitude ratio of torsional motion to vertical motion and phase difference satisfy: The solution yields: in: and These represent the vertical coordinates and torsional coordinates of the vertical mode, respectively. The amplitude ratio of vertical motion to torsional motion and phase difference satisfy: The solution yields: in: and These represent the vertical and torsional coordinates of the torsional mode, respectively. This refers to the width of the bridge deck. For the self-integration of the vertical mode; This is the cross-integral of the vertical and torsional modes.

8. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 1, characterized in that: In step four, when the structural frequencies of the vertical and torsional modes are the same, the modal frequencies are obtained using the relationship between eigenvalues, frequencies, and damping. Damping ratio for: in: and These represent the frequency and damping of the vertical and torsional modes, respectively. The frequency of motion; To calculate the imaginary value in a complex number; To calculate the real value of a complex number; The similarity coefficient between the vertical and torsional modal shapes; and These are the single-degree-of-freedom damping ratios in the vertical and torsional directions, respectively, caused by uncoupled self-excited forces; and These are the frequencies in the vertical and torsional directions caused by the uncoupled self-excited force, respectively. and For dimensionless mass and moment of inertia; and For flutter derivative, ; Amplitude ratio of torsional mode to vertical mode and phase difference satisfy: The solution yields: Amplitude ratio of vertical mode to torsional mode and phase difference satisfy: The solution yields: in: and It is a constant related to the initial displacement and velocity.

9. The improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter in long flexible structures according to claim 1, characterized in that: Step five involves determining the critical flutter speed: 51) Determine whether the vertical and torsional frequencies of the initial structure are the same, select the corresponding explicit analytical solution formula, and solve the modal frequencies through iterative calculation to obtain the solution values ​​of the modal frequencies in each direction; 52) After the modal frequencies converge, the damping ratio, amplitude ratio and phase difference of each mode are directly calculated according to the explicit analytical form given in step four. 53) Gradually increase the incoming wind speed and calculate the modal damping ratio of the structure in the vertical and torsional directions respectively. When the wind speed increases to a certain value, if the modal damping in any direction changes from positive to negative, it indicates that the system has changed from damping energy dissipation to continuously absorbing external energy, and the structure will undergo self-excited vibration. The wind speed corresponding to this time is the flutter critical wind speed.

10. A system for implementing the improved canonical perturbation high-order expansion prediction method for the critical wind speed of bending-torsional coupling flutter of long flexible structures as described in any one of claims 1-9, characterized in that: include: The data acquisition module is used to design and execute segmental model wind tunnel tests based on the Scanlan linear self-excited force model, and to collect the structural physical parameters and aerodynamic characteristic parameters of the bridge. The dynamic modeling module is used to construct the bridge motion control matrix equations based on modal displacements, complete the small parameter perturbation expansion of the equations, and output the perturbation differential equations of each order. The analytical solution module is used to derive explicit analytical solutions for modal frequencies, damping ratios, amplitude ratios, and phase differences by eliminating long-term terms and considering both equal and unequal frequencies, solving eigenvalues ​​and modal vectors step by step. The wind speed prediction module is used to select the corresponding analytical solution set based on the initial frequency relationship, directly solve the damping ratio after iteratively calculating the modal frequency, and determine the flutter critical wind speed based on the variation law of the damping ratio with wind speed.

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