A method and system for predicting the flutter critical wind speed of a long-span bridge based on an L-P perturbation method

Abstract

This invention relates to a method for predicting the critical flutter wind speed of long-span bridges based on the L-P perturbation method, comprising the following steps: S1, obtaining the physical characteristics and aerodynamic parameters of the bridge as basic data through wind tunnel tests; S2, establishing the bridge motion control matrix equation based on modal displacement; S3, deriving explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch based on the L-P perturbation method; S4, iteratively solving the modal frequencies, and directly solving the damping ratio, amplitude ratio, and phase difference on this basis, observing the change of the modal damping ratio ξ with the wind speed U in each direction, and determining the critical flutter wind speed; solving the problem that the existing explicit closed-form solution of the three-degree-of-freedom flutter initiation condition requires multiple iterative calculations of the modal frequencies, damping ratios, and mode shapes, and is difficult to clearly explain the influence of structural and aerodynamic parameters on aerodynamic instability, thus failing to accurately predict the critical flutter wind speed of long-span bridges.

Description

Technical Field

[0001] This invention belongs to the field of wind-resistant design and aeroelasticity technology of civil engineering structures. It relates to a method and system for predicting the critical wind speed of flutter in long-span bridges based on the LP perturbation method, which is applicable to the aerodynamic stability analysis of flutter in long-span bridge structures. Background Technology

[0002] Long-span bridge structures are particularly sensitive to wind loads. The self-excited forces in wind loads can introduce negative damping effects into the bridge system, potentially leading to aerodynamic instability. Flutter, as a typical wind-induced coupled vibration phenomenon, requires a thorough understanding of its mechanism for wind-resistant bridge design.

[0003] Complex eigenvalue analysis is an effective method for determining the critical flutter wind speed of bridges and providing important information such as modal frequencies and damping ratios. However, this method has limitations in explaining how structural and aerodynamic parameters affect flutter initiation. Especially in the case of three-degree-of-freedom aerodynamic instability, particularly under strong coupling, the specific contributions of aerodynamic derivatives and structural characteristics remain unclear, sometimes revealing new physical properties. While existing explicit closed-form solutions for three-degree-of-freedom galloping and flutter initiation conditions can accurately quantify the influence of each parameter on the initial wind speed, the calculation requires multiple iterations of modal frequencies, damping ratios, and mode shapes, increasing computational complexity.

[0004] Conventional perturbation methods, by utilizing the partial solvability of the system and small perturbations, can provide explicit closed-form solutions for modal damping and frequency. However, when dealing with strongly coupled systems, it is essential to comprehensively classify specific parameters and adjust the distribution of matrices of each order to ensure that specific terms are clearly defined and interpretable. Therefore, how to rapidly and accurately obtain closed-form solutions for aerodynamic instability initiation conditions under strongly coupled environments, and clearly reveal the specific influence of each coupling component on the instability mechanism, is a technical problem that needs to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, in order to solve the problem that the explicit closed solution of the initiation condition of existing three-degree-of-freedom flutter requires multiple iterative calculations of modal frequency, damping ratio and mode shape, and is difficult to clearly explain the influence of structural parameters and aerodynamic parameters on aerodynamic instability, and cannot predict the critical flutter wind speed of long-span bridges well, this invention provides a method and system for predicting the critical flutter wind speed of long-span bridges based on the LP perturbation method.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method for predicting the critical flutter wind speed of long-span bridges based on the LP perturbation method includes the following steps:

[0008] S1. Obtain the physical properties and aerodynamic parameters of the bridge as basic data through wind tunnel testing;

[0009] S2. Establish the bridge motion control matrix equation based on modal displacement, which includes: establishing the dynamic displacement expression of the bridge structure and using modal expansion; constructing a unit-length self-excited aerodynamic dynamic expression linearized around the static displacement position, and obtaining the corresponding generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing generalized mass, damping, and stiffness matrices.

[0010] S3. Based on the LP perturbation method, derive the explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch, including: defining new variables and transforming their differential forms, rewriting the governing equations; constructing the perturbation expansion framework; performing power series expansion on the eigenvalues ​​and displacement solutions based on the LP perturbation method; solving for the eigenvalues ​​corresponding to the zeroth, first, and second orders in sequence; and substituting the eigenvalues ​​of each order into the equations containing the small parameter ε to solve for the explicit solutions for the frequencies and damping ratios of each modal branch.

[0011] S4. Iteratively solve for the modal frequencies, and then directly solve for the damping ratio, amplitude ratio, and phase difference. Observe the change of the modal damping ratio ξ with the wind speed U in each direction to determine the critical wind speed for flutter.

[0012] A flutter critical wind speed prediction system for long-span bridges based on the LP perturbation method, applied to the aforementioned flutter critical wind speed prediction method for long-span bridges, includes:

[0013] The data acquisition module is used to obtain the bridge's physical characteristics and aerodynamic parameters, including flutter derivative, bridge width B, mass per unit length m, moment of inertia I, and frequencies in the vertical, lateral, and torsional directions. Damping ratio ;

[0014] The dynamic model module is used to establish the bridge motion control matrix equation based on modal displacement. It includes: establishing the dynamic displacement expression of the bridge structure and expanding it using modal expansion; constructing a self-excited aerodynamic expression linearized around the static displacement position and obtaining the corresponding generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing generalized mass, damping, and stiffness matrices.

[0015] The perturbation method solution module uses the LP perturbation method to derive explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch. This process includes: defining new variables and transforming their differential form to rewrite the governing equations; constructing the perturbation expansion framework; performing power series expansions on the eigenvalues ​​and displacement solutions based on the LP perturbation method; sequentially solving for the eigenvalues ​​corresponding to the zeroth, first, and second orders; and substituting the eigenvalues ​​into equations containing a small parameter ε to solve for the explicit solutions for the frequencies and damping ratios of each modal branch.

[0016] The wind speed prediction module iteratively solves for the modal frequencies and directly solves for the damping ratio, amplitude ratio, and phase difference. By observing the change of the modal damping ratio ξ with the wind speed U, the critical wind speed for flutter is determined.

[0017] The beneficial effects of this invention are as follows:

[0018] The flutter critical wind speed prediction method for long-span bridges based on the LP perturbation method disclosed in this invention has the following advantages:

[0019] 1) Improved computational efficiency:

[0020] By simply iterating over the modal frequencies, the modal damping ratio and the corresponding eigenvectors can be obtained directly from the calculated modal frequencies, simplifying the calculation process, avoiding complex iterative calculations, and significantly improving computational efficiency.

[0021] 2) Enhanced explanation of the mechanism:

[0022] Decomposing complex coupled nonlinear problems into a linear superposition of structural, uncoupled, and coupled components helps to clearly reveal the influence of structural and aerodynamic parameters on aerodynamic instability, thus clearly demonstrating the impact of aerodynamic coefficients and structural parameters on aerodynamic instability.

[0023] 3) Improved prediction accuracy:

[0024] It can quickly and accurately obtain the closed-loop solution of the aerodynamic instability initiation condition and clearly reveal the influence of each coupled component on the instability mechanism, thereby improving the accuracy of predicting the critical wind speed for flutter in long-span bridges.

[0025] 4) Wide applicability:

[0026] It is applicable to the aerodynamic stability analysis of flutter in long-span bridge structures, especially in strongly coupled environments, and can handle three-degree-of-freedom (3DOF) aerodynamic instability problems, with wide applicability.

[0027] 5) Completeness of the technical solution:

[0028] It not only provides a prediction method, but also proposes a corresponding system implementation scheme, including a data acquisition module, a dynamic model module, a perturbation method solution module, and a wind speed prediction module, forming a complete technical solution.

[0029] 6) Scientific evidence provided:

[0030] It can provide a scientific basis for the wind-resistant design and operation and maintenance of long-span bridges, and help prevent and reduce accidents caused by bridge flutter, thereby reducing economic losses.

[0031] 7) Coupling effect analysis:

[0032] It can clearly explain the impact of coupling effects, and is expressed by a linear combination of total damping, providing a clear insight into the coupling mechanism and helping to understand the coupling effects in different modal branches.

[0033] 8) Simplify the analysis process:

[0034] For bridge flutter analysis with large frequency intervals, the method of this invention can ignore the influence of coupling components on frequency, thereby simplifying the flutter analysis process and only considering the influence of coupling terms on the damping ratio.

[0035] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0036] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0037] Figure 1 This is a flowchart of the flutter critical wind speed prediction method for long-span bridges based on the LP perturbation method of the present invention.

[0038] Figure 2 For the present invention Figure 1 Step S2 shows the aerodynamic diagram of the bridge cross section;

[0039] Figure 3 This is a cross-sectional view of the Suramadu Bridge in an embodiment of the present invention;

[0040] Figure 4 This refers to the flutter derivative of the bridge deck section of the Suramadu Bridge in this embodiment of the invention; wherein... Figure 4 (a) and (d) show the variation of flutter derivatives related to vertical motion with wind speed. Figure 4(b) and (c) show the variation of flutter derivatives related to torsional motion with wind speed. Figure 4 (e) and (f) show the variation of flutter derivatives related to lateral motion with wind speed;

[0041] Figure 5 This is a comparison between the eigenvalue analysis in the embodiments of the present invention and the three-dimensional analytical solution proposed in the present invention; wherein Figure 5 (a) Comparison of the modal frequencies of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. Figure 5 (b) Comparison of the modal damping ratios of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. Figure 5 (c) Comparison of the modal amplitude ratios of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. The left figure, from top to bottom, shows vertical / torsional, lateral / vertical, and vertical / lateral; the right figure, from top to bottom, shows torsion / vertical, torsion / lateral, and lateral / torsional. Figure 5 (d) Comparison of the modal phase difference of the vertical, lateral and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculation. The left figure shows the lateral / vertical, vertical / torsional and vertical / lateral modal branches from top to bottom, and the right figure shows the torsional / lateral, lateral / torsional and torsional / vertical modal branches from top to bottom.

[0042] Figure 6 This refers to the contribution of each component to the total damping in the embodiments of the present invention; wherein Figure 6 (a) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the vertical modal branch; Figure 6 (b) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the torsional mode branch;

[0043] Figure 7 The contributions of each item in the embodiments of the present invention to the coupling damping are shown below; wherein Figure 7 (a) represents the contribution of the transverse coupling term to the coupling damping in the vertical modal branch; Figure 7 (b) represents the contribution of the torsional coupling term to the coupling damping in the vertical modal branch; Figure 7 (c) represents the contribution of the transverse coupling term to the coupling damping in the torsional modal branch; Figure 7 (d) represents the contribution of the vertical coupling term to the coupling damping in the torsional mode branch. Detailed Implementation

[0044] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0045] like Figure 1 The method for predicting the critical flutter wind speed of long-span bridges based on the LP perturbation method, as shown, includes the following steps:

[0046] S1. Collect basic bridge data;

[0047] Specifically:

[0048] Flutter derivative and bridge width determined by wind tunnel testing B Mass per unit length m Moment of inertia I Vertical frequency Vertical damping ratio Horizontal frequency Lateral damping ratio Frequency of twisting direction Torsional damping ratio .

[0049] S2. Establish the bridge motion control matrix equation based on modal displacement;

[0050] The specific steps are as follows:

[0051] S21. Establish the expression for the dynamic displacement of the bridge: the bridge deck at time... The dynamic displacements relative to the static displacement position in the vertical, lateral, and torsional directions are denoted as follows: , and These dynamic displacements are represented by their respective fundamental modes, as shown in the following expressions:

[0052]

[0053] in, , and It is the mode shape; Modal displacement; For the cross-directional position.

[0054] S22. Establish the expression for the self-excited force per unit length: (e.g.) Figure 2 As shown, the self-excited forces per unit length linearized around the static displacement position, namely lift (vertical), drag (lateral), and torsional moment, are expressed as follows:

[0055]

[0056] in, air density; Average wind speed; This refers to the width of the bridge deck. To reduce the frequency; The frequency of motion; , and ( The aerodynamic derivative or flutter derivative is a function of the reduced frequency; The self-excited lift per unit length; The self-excited resistance per unit length; The self-excited torsional moment per unit length.

[0057] S23. Establish the expression for the generalized modal force: Integrate the self-excited force per unit length in step S22 to obtain the generalized modal force, as shown in the following expression:

[0058]

[0059] S24. Establish the bridge motion control matrix equation based on modal displacement:

[0060]

[0061] in, , and These are the generalized mass matrix, damping matrix, and stiffness matrix, respectively. and ( For modal mass, damping ratio, and frequency; and These are the generalized displacement vector and force vector; and These are the aerodynamic stiffness matrix and the aerodynamic damping matrix; For modal integrals.

[0062] Based on this, the control matrix equation can also be further expressed as:

[0063]

[0064] in, and These are the uncoupled damping ratio and frequency caused by the uncoupled self-excited force, respectively. , and Let be the dimensionless mass and the moment of inertia.

[0065] S3. Based on the LP perturbation method, derive explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch.

[0066] The specific steps are as follows:

[0067] S31. Define a new variable and transform the differential form: Define a new independent variable. By applying the chain rule of differentials, the differential with respect to time is transformed into a partial differential with respect to the eigenvalues ​​of each modal branch:

[0068]

[0069] S32. Rewrite the governing equation: Equation (5a) in step S24 can be replaced with the new independent variable defined in step S31. Rewritten as:

[0070]

[0071] in, , and Let the mass matrix, damping matrix, and stiffness matrix of a second-order damped linear oscillator be represented, respectively. and They represent right The first and second derivatives.

[0072] Solution vector It should be represented as a linear combination of modal vectors:

[0073]

[0074] in, This represents the modal vector of each branch.

[0075] S33. Constructing the perturbation expansion framework: To ensure the initial linearity of equation (7a) in step S32 and the feasibility of each order of iteration, the mass matrix, damping matrix, and stiffness matrix can be expressed as the sum of the zeroth-order matrix and the first-order matrix, where... The small perturbation parameter of the nonlinear term in the equation is, i.e. The specific expression is:

[0076]

[0077] S34. Based on the LP perturbation method, perform a power series expansion of the eigenvalues ​​and displacement solutions: assuming eigenvalues reconciliation Press The power series expansion is as follows:

[0078]

[0079] S35. Solve for the zeroth-order term: Substitute the expansion from step 34 into the rewritten governing equation (7a) from step S32, and collect... Terms of the same order. Assume the equation is valid for all sufficiently small terms. If established, then each The coefficient of a power should be zero. Therefore, for the zeroth-order term ( ),untie For the unperturbed or uncoupled terms, the eigenvalues ​​are:

[0080]

[0081] in, ( ) is a constant related to the initial displacement and velocity.

[0082] S36. Solve the equation for the first-order term: For the first-order term ( ):

[0083]

[0084] Substituting the zeroth-order solution from step S35 into equation (11), and then setting the coefficients of each long-term term to zero, we obtain the corresponding first-order eigenvalues ​​and first-order solutions as follows:

[0085]

[0086] When solving for a particular solution with non-identical eigenvalues, the coefficients obtained from the homogeneous equation terms are: ;

[0087] S37. Solve the second-order term equation: For the second-order term ( ):

[0088]

[0089] Since the first-order solution obtained in step S36 has a periodic solution, substituting it into equation (13) yields the following equation:

[0090]

[0091] The corresponding eigenvalues ​​and solutions are:

[0092]

[0093] in, The dimensionless coefficients are obtained through the chain rule in the differentiation process.

[0094] S38. Calculate the modal frequency, damping ratio, amplitude ratio, and phase difference of each modal branch: Substitute the eigenvalues ​​obtained in steps S35-S37 into the step containing the small parameter in step S34. Equation (9b), by assuming The solution is periodic, and the complex modal branch frequency is... Damping ratio Through eigenvalues By solving this problem, the explicit solutions for the frequency and damping ratio of each modal branch can be obtained.

[0095] For vertical modal branching, and The solutions for its modal frequencies and damping ratios are expressed as follows:

[0096]

[0097] in, , , ,and This represents the similarity factor between different mode shapes. Specifically, the frequencies of the coupled modal branches can be approximated by the frequencies affected by the uncoupled self-excited forces, i.e. and .

[0098] Vertical mode vector As shown below,

[0099]

[0100] The amplitude ratio of lateral and torsional motion relative to vertical motion ( ) and phase difference ( ) are respectively and Without considering higher-order terms, they can be simplified to:

[0101]

[0102] For the transverse modal branch, the solution for the relevant parameters is expressed as follows:

[0103]

[0104] For the torsional modal branch, the solution for the relevant parameters is expressed as follows:

[0105]

[0106] S4. Iteratively solve for the modal frequencies, and then directly solve for the damping ratio, amplitude ratio, and phase difference to determine the critical flutter wind speed.

[0107] The specific steps are as follows:

[0108] S41. Iterative solution of modal frequencies: The terms on the right side of the modal frequencies in each modal branch in step S38 contain unknown frequencies, so the solution of modal frequencies can be obtained through iterative calculation.

[0109] S42. Solve for damping ratio, amplitude ratio and phase difference: After completing the modal frequency solution in step S41, based on the form of the solution for damping ratio, amplitude ratio and phase difference of each modal branch in step S38, it is clear that damping ratio, amplitude ratio and phase difference can be solved directly without iteration.

[0110] S43. Determine the critical flutter wind speed: Observe the modal damping ratios in the vertical, horizontal, and torsional directions calculated in step S42. With wind speed By gradually increasing the damping, the flutter initiation wind speed can be determined when the damping of a certain mode becomes negative.

[0111] Example

[0112] Take the flutter problem of the Suramadou Bridge as an example. This bridge is a cable-stayed bridge with a main span of 434 meters, and adopts... Figure 3 The composite beam section shown is relatively steep. Figure 4 The flutter derivative, determined through wind tunnel testing of a segmental model, is shown. Model width. B =6 meters, mass per unit length m =9.64 Moment of inertia I =0.407 The frequencies and damping ratios in the vertical, horizontal, and torsional directions are as follows: Hz, Hz, Hz; , , .in Figure 4 (a) and (d) show the variation of flutter derivatives related to vertical motion with wind speed. Figure 4 (b) and (c) show the variation of flutter derivatives related to torsional motion with wind speed. Figure 4 (e) and (f) show the variation of flutter derivatives related to lateral motion with wind speed;

[0113] Figure 5 The diagram compares the modal frequencies, damping ratios, amplitude ratios, and phase differences calculated using eigenvalue analysis with those obtained from the proposed three-dimensional analytical solution. Figure 5 (a) Comparison of the modal frequencies of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. Figure 5(b) Comparison of the modal damping ratios of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. Figure 5 (c) Comparison of the modal amplitude ratios of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. The left figure, from top to bottom, shows vertical / torsional, lateral / vertical, and vertical / lateral; the right figure, from top to bottom, shows torsion / vertical, torsion / lateral, and lateral / torsional. Figure 5 (d) Comparison of the modal phase differences of the vertical, lateral, and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculations. The left figure, from top to bottom, shows lateral / vertical, vertical / torsional, and vertical / lateral; the right figure, from top to bottom, shows torsional / lateral, lateral / torsional, and torsional / vertical. Clearly, the results are completely consistent within the wind speed range of 0 to 12 m / s. It should be emphasized that, due to the assumption of relatively small damping, the proposed three-dimensional analytical solution has sufficient accuracy near the critical wind speed region. However, as the absolute value of the damping ratio increases, the results predicted by this method will deviate from the results of eigenvalue analysis. The lateral modal frequency and damping ratio remain almost constant, and therefore will not be discussed further. The torsional modal frequency and damping ratio decrease with increasing wind speed, while the vertical modal frequency and damping ratio increase with increasing wind speed. When the wind speed is 9.87 m / s, the torsional modal branch exhibits negative damping.

[0114] According to the analytical solution of the modal damping ratio, the total damping ratio can be divided into three components: structural damping, uncoupled damping, and coupled damping. Figure 6 The contribution of each component in the vertical and torsional mode branches to the total damping is shown at different wind speeds. Figure 6 (a) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the vertical modal branch; Figure 6 (b) shows the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the torsional modal branch; here, it is assumed that the structural damping is constant. For the vertical modal branch, the increase in total damping mainly comes from uncoupled damping. The contribution. Within the calculated wind speed range, Since the value is negative, the uncoupled damping is positive. The coupled damping from the lateral and torsional mode branches is also positive and increases with increasing wind speed. For the torsional mode branch, uncoupled damping initially dominates, but the coupled term eventually dominates the total damping. The trend of uncoupled damping is similar to... The changes are consistent. The coupled damping is negative and gradually increases with increasing wind speed. At the critical wind speed, although both structural damping and uncoupled damping are positive, the coupled damping is sufficiently negative, leading to aerodynamic instability.

[0115] The proposed analytical solution can also clearly explain the effect of coupled damping, since the total damping is expressed as a linear combination of the damping terms. As can be seen from equations (15b) and (17b), coupled damping contains two components from two other modal branches, and each component further contains four different terms labeled "terms 1-4" from left to right. Figure 7 The percentage contribution of each term in the vertical and torsional mode branches to the total coupled damping is shown at different wind speeds, providing a clear insight into the coupling mechanism. Among them, Figure 7 (a) represents the contribution of the transverse coupling term to the coupling damping in the vertical modal branch; Figure 7 (b) represents the contribution of the torsional coupling term to the coupling damping in the vertical modal branch; Figure 7 (c) represents the contribution of the transverse coupling term to the coupling damping in the torsional modal branch; Figure 7 (d) represents the contribution of the vertical coupling term to the coupling damping in the torsional modal branch;

[0116] In the vertical modal branch, lateral coupling damping is significant only at relatively low wind speeds. As wind speed increases, the coupling effect is mainly dominated by the torsional coupling component. In the torsional coupling damping term, term 1 in equation (15b) (i.e. The vertical coupling component becomes the dominant factor. In the torsional modal branch, the coupling effect is mainly dominated by the vertical coupling component, especially term 2. Therefore, the coupling negative damping in the torsional modal branch is also essentially caused by the vertical coupling component in equation (17b). The decision is made. The solution for the damping ratio can be further simplified by retaining only the most important terms in the coupled damping components. A comparison of the simplified solutions can be found in [link to simplified solution]. Figure 5 (b). Clearly, the simplified results have sufficient accuracy for both the vertical and torsional modal branches.

[0117] For a given frequency The coupling effect exhibits opposite behavior in different modal branches. For example, vertical-bending-torsional coupling damping produces positive damping in the vertical modal branch but negative damping in the torsional modal branch. Furthermore, these coupling effects are frequency-dependent on the target modal branch, explaining why the coupling effect is more pronounced in the torsional modal branch. The coupling effect is also affected by the frequency difference. The effect of coupling damping is enhanced when frequencies are close to each other; the coupling effect weakens as the frequency difference increases. It is worth noting that when... At that time, the negative coupling damping originally observed in the torsional direction will turn into positive damping.

[0118] Figure 5(a) shows the contribution of the coupling component to the total frequency. The results indicate that the effect of the coupling component on the total frequency is negligible. The damping ratio calculated using the frequency without the coupling component is the same as the damping ratio calculated using the exact frequency. Therefore, for flutter analysis of bridges with large frequency intervals, the effect of the coupling component on the frequency can be ignored. Flutter analysis can be simplified to considering only the effect of the coupling term on the damping ratio.

[0119] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for predicting the critical flutter wind speed of long-span bridges based on the LP perturbation method, characterized in that, Includes the following steps: S1. Obtain the physical properties and aerodynamic parameters of the bridge as basic data through wind tunnel testing; S2. Establish the bridge motion control matrix equation based on modal displacement, which includes: establishing the dynamic displacement expression of the bridge structure and using modal expansion; constructing a unit-length self-excited aerodynamic expression linearized around the static displacement position, and obtaining the corresponding generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing generalized mass, damping, and stiffness matrices. The equations for the bridge motion control matrix based on modal displacement are established as follows: in, These are the generalized mass matrix, damping matrix, and stiffness matrix, respectively. For modal mass, damping ratio, and frequency; These are the generalized displacement vector and force vector; These are the aerodynamic stiffness matrix and the aerodynamic damping matrix; For modal integrals; air density; Average wind speed; To reduce the frequency; The frequency of motion; The aerodynamic derivative or flutter derivative is a function of the reduced frequency; The control matrix equation (4a) is further expressed as: in, These are the uncoupled damping ratio and frequency caused by the uncoupled self-excited force, respectively. For dimensionless mass and moment of inertia; For the width of the bridge, It is the moment of inertia; S3. Derive the explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch based on the LP perturbation method, including: defining new variables and transforming their differential form, rewriting the governing equations; constructing the perturbation expansion framework; performing power series expansions on the eigenvalues ​​and displacement solutions based on the LP perturbation method; solving for the eigenvalues ​​corresponding to the zeroth, first, and second orders in sequence; and substituting the eigenvalues ​​of each order into the system containing small parameters. The equations are used to find explicit solutions for the frequency and damping ratio of each modal branch; Step S3 specifically includes: S31. Define a new variable and transform the differential form: Define a new independent variable. By applying the chain rule of differentials, the differential with respect to time is transformed into a partial differential with respect to the eigenvalues ​​of each modal branch: S32. Rewrite the governing equation: Equation (5a) in step S24 is modified using the new independent variable defined in step S31. Rewritten as: in, Let the mass matrix, damping matrix, and stiffness matrix of a second-order damped linear oscillator be represented, respectively. They represent The first and second derivatives; Solution vector It should be represented as a linear combination of modal vectors: in Represents the modal vectors of each branch; S33. Constructing the perturbation expansion framework: To ensure the initial linearity of equation (7a) in step S32 and the feasibility of each order of iteration, the mass matrix, damping matrix, and stiffness matrix can be expressed as the sum of the zeroth-order matrix and the first-order matrix, where... The small perturbation parameter of the nonlinear term in the equation is, i.e. The specific expression is: S34. Based on the LP perturbation method, perform a power series expansion of the eigenvalues ​​and displacement solutions: assuming eigenvalues reconciliation The power series expansion is: ； S4. Iteratively solve for the modal frequencies, and then directly solve for the damping ratio, amplitude ratio, and phase difference. Observe the modal damping ratio in each direction. With wind speed The changes in wind speed were used to determine the critical flutter speed.

2. The method for predicting the critical flutter wind speed of long-span bridges as described in claim 1, characterized in that, The basic bridge data collected in step S1 includes: flutter derivative determined by wind tunnel testing, mass per unit length, etc. Vertical frequency Vertical damping ratio Horizontal frequency Lateral damping ratio Frequency of twisting direction Torsional damping ratio .

3. The method for predicting the critical flutter wind speed of long-span bridges as described in claim 2, characterized in that, Step S2 is as follows: S21. Establish the expression for the dynamic displacement of the bridge: the bridge deck at time... The dynamic displacements relative to the static displacement position in the vertical, lateral, and torsional directions are denoted as follows: The expressions for each of their basic modes are as follows: in It is the mode shape; For modal displacement; For the cross-directional position; S22. Establish the expression for the self-excited force per unit length: Based on quasi-steady-state theory, the self-excited force per unit length linearized around the static displacement position, namely lift, drag, and pitching moment, is expressed as follows: in, The self-excited lift per unit length; The self-excited resistance per unit length; The self-excited torsional moment per unit length; S23. Establish the expression for the generalized modal force: Integrate the self-excited force per unit length in step S22 to obtain the generalized modal force, as shown in the following expression: in, For vertical generalized modal forces; It is a transverse generalized modal force; To reverse the generalized modal force.

4. The method for predicting the critical flutter wind speed of long-span bridges as described in claim 1, characterized in that, Step S3 also includes: S35. Solve for the zeroth-order term: Substitute the expansion from step 34 into the rewritten governing equation (7a) from step S32, and collect... Terms of the same order, assuming the equation is for all sufficiently small If established, then each The coefficient of a power should be zero, therefore for the zeroth-order term... ,untie For the unperturbed or uncoupled terms, the eigenvalues ​​are: in, It is a constant related to the initial displacement and velocity; S36. Solve the first-order term equation: For the first-order term... : Substituting the zeroth-order solution from step S35 into equation (11), and then setting the coefficients of each long-term term to zero, we obtain the corresponding first-order eigenvalues ​​and first-order solutions as follows: When solving for a particular solution with non-identical eigenvalues, the coefficients obtained from the homogeneous equation terms are: ; S37. Solve the second-order term equation: For the second-order term... : Since the first-order solution obtained in step S36 has a periodic solution, substituting it into equation (13) yields the following equation: The corresponding eigenvalues ​​and solutions are: in, These are the dimensionless coefficients obtained through the chain rule in the differentiation process; S38. Calculate the modal frequency, damping ratio, amplitude ratio, and phase difference of each modal branch: Substitute the eigenvalues ​​obtained in steps S35-S37 into the step containing the small parameter in step S34. Equation (9b), by assuming The solution is periodic, and the complex modal branch frequency is... Damping ratio Through eigenvalues By solving this problem, the explicit solutions for the frequency and damping ratio of each modal branch can be obtained.

5. The method for predicting the critical flutter wind speed of long-span bridges as described in claim 4, characterized in that, In step S38: For vertical modal branching, The solutions for its modal frequencies and damping ratios are expressed as follows: in, and This represents the similarity factor between different mode shapes; specifically, the frequency of the coupled modal branch can be approximated by the frequency affected by the uncoupled self-excited force, i.e. ; Vertical mode vector As shown below, The amplitude ratio of lateral and torsional motion relative to vertical motion and phase difference They are respectively Without considering higher-order terms, it simplifies to: For the transverse modal branch, the solution for the relevant parameters is expressed as follows: For the torsional modal branch, the solution for the relevant parameters is expressed as follows: 。 6. The method for predicting the critical flutter wind speed of long-span bridges as described in claim 5, characterized in that, Step S4 specifically involves iteratively solving for the modal frequencies of each branch in step S38, and then directly calculating the damping ratio, amplitude ratio, and phase difference of each modal branch. The calculated modal damping ratios in the vertical, lateral, and torsional directions are then observed. With wind speed By gradually increasing the damping, the flutter initiation wind speed can be determined when the damping of a certain mode becomes negative.

7. A flutter critical wind speed prediction system for long-span bridges based on the LP perturbation method, applied to the flutter critical wind speed prediction method for long-span bridges as described in any one of claims 1 to 6, characterized in that, include: The data acquisition module is used to obtain the bridge's physical properties and aerodynamic parameters, including flutter derivative and bridge width. Mass per unit length Moment of inertia and frequencies in the vertical, horizontal and torsional directions. Damping ratio ; The dynamic model module is used to establish the bridge motion control matrix equation based on modal displacement. It includes: establishing the dynamic displacement expression of the bridge structure and expanding it using modal expansion; constructing a self-excited aerodynamic expression linearized around the static displacement position and obtaining the corresponding generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing generalized mass, damping, and stiffness matrices. The perturbation method solution module derives explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch using the LP perturbation method. This process includes: defining new variables and transforming their differential form to rewrite the governing equations; constructing the perturbation expansion framework; performing power series expansions on the eigenvalues ​​and displacement solutions based on the LP perturbation method; sequentially solving for the eigenvalues ​​corresponding to the zeroth, first, and second orders; and substituting the eigenvalues ​​of each order into the solution containing small parameters. The equations are used to find explicit solutions for the frequency and damping ratio of each modal branch; The wind speed prediction module iteratively solves for the modal frequencies and directly calculates the damping ratio, amplitude ratio, and phase difference based on these results. The modal damping ratio is observed during the calculation. The critical flutter wind speed is determined by varying the wind speed U.

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