A method and system for predicting the critical wind speed of power line galloping based on L-P perturbation method

Abstract

The present application relates to a kind of transmission line galloping critical wind speed prediction method based on L-P perturbation method, comprising the following steps: S1, the physical characteristics and aerodynamic parameters of transmission line are obtained by wind tunnel test as basic data;S2, establish the motion control matrix equation of transmission line based on modal displacement;S3, explicit solution of modal frequency, damping ratio and modal characteristics of each modal branch is derived based on L-P perturbation method;S4, iteratively solve modal frequency, on this basis directly solve damping ratio, amplitude ratio and phase difference, observe the change of modal damping ratio ξ in each direction with wind speed U, determine the critical wind speed of galloping;The explicit closed solution of the existing three-degree-of-freedom galloping start-up condition needs to be calculated multiple times iteratively on modal frequency, damping ratio and mode shape, and it is difficult to clearly explain the influence of structural parameters and aerodynamic parameters on aerodynamic instability, and the critical wind speed of transmission line galloping cannot be well predicted.

Description

Technical Field

[0001] This invention belongs to the field of wind-resistant design and aeroelasticity technology of civil engineering structures, and relates to a method and system for predicting the critical wind speed of transmission line galloping based on the LP perturbation method, which is applicable to the aerodynamic stability analysis of transmission line structure galloping. Background Technology

[0002] Galloping of transmission lines refers to a low-frequency (approximately 0.1–3 Hz), large-amplitude (approximately 5–300 times the conductor diameter) self-excited vibration phenomenon that occurs when wind acts on transmission conductors that have become non-circular due to icing under certain conditions. During galloping, the peak vibration value can typically reach over 10 meters, easily causing phase-to-phase flashovers, hardware damage, line tripping and power outages, or serious accidents such as conductor burns, tower collapses, and conductor breakage, resulting in significant economic losses.

[0003] Regarding the conductor galloping mechanism of transmission lines, there are Den Hartog's vertical galloping mechanism, O. Nigol's torsional galloping mechanism from Canada, and the torsional feedback mechanism from Japan. In terms of mathematical models, there is WN McDaniel's linear model of single-conductor galloping from the United States, solved using dynamic linear stability theory; Otsuki et al. from Japan applied the energy balance method to perform approximate nonlinear analysis on the single-conductor case; Blevins and Iwan (1974) and Yu et al. (1991, 1992)'s two-degree-of-freedom pitch and torsion models; P. Yu et al. (1994)'s three-degree-of-freedom model (using the perturbation method to obtain an analytical solution); and YM Desai et al. (1996)'s three-degree-of-freedom model (solved using the finite element method), etc.

[0004] As a flexible structure, power transmission lines are highly sensitive to wind loads and easily affected by self-excited forces, which can introduce negative damping into the system and lead to aerodynamic instability. Galloping is a typical type of wind-induced coupled vibration. Therefore, a thorough investigation of the mechanisms of aerodynamic instability is crucial for the wind-resistant design of power transmission lines.

[0005] Currently, researchers are striving to deepen their understanding of three-degree-of-freedom (3DOF) aerodynamic instability through closed-form solutions. However, for 3DOF aerodynamic instability, especially in strongly coupled cases, the contributions of aerodynamic derivatives and structural properties to instability remain unclear and may exhibit new physical characteristics. Existing explicit closed-form solutions to the 3DOF galloping initiation conditions, while capable of accurately quantifying the influence of each parameter on the initial wind speed, require multiple iterative calculations of modal frequencies, damping ratios, and mode shapes. In contrast, conventional perturbation methods, by utilizing solvable system components and small perturbations, can provide explicit closed-form solutions for modal damping and frequencies.

[0006] However, in strongly coupled systems, a comprehensive classification of specific parameters is essential, and the distribution of matrices of each order needs to be adjusted to ensure that specific terms are clearly definable and interpretable. Therefore, how to quickly and accurately obtain the closed-form solution of the aerodynamic instability initiation condition under strongly coupled conditions, clearly reveal the influence of each coupling component on the instability mechanism, and thus accurately predict the critical wind speed for power transmission line galloping, is a key technical problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0007] In view of this, in order to solve the problem that the explicit closed solution of the vibration initiation condition of the existing three-degree-of-freedom galloping requires multiple iterative calculations of the 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 wind speed of transmission line galloping well, the present invention provides a method and system for predicting the critical wind speed of transmission line galloping based on the LP perturbation method.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] A method for predicting the critical wind speed for power line galloping based on the LP perturbation method includes the following steps:

[0010] S1. Obtain the physical properties and aerodynamic parameters of the transmission line as basic data through wind tunnel testing;

[0011] S2. Establish the motion control matrix equation for the transmission line based on modal displacement, which includes: establishing the dynamic displacement expression of the transmission line structure and expanding it using modal expansion; constructing the expression of the self-excited aerodynamic force of the icing transmission line based on quasi-steady-state theory; simplifying the aerodynamic force expression through a linear force model and obtaining the generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing the generalized mass, damping, and stiffness matrices.

[0012] 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.

[0013] 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 galloping.

[0014] A critical wind speed prediction system for power transmission line galloping based on the LP perturbation method, applied to the aforementioned critical wind speed prediction method for power transmission line galloping, includes:

[0015] The data acquisition module is used to acquire the physical characteristics and aerodynamic parameters of the transmission line, including aerodynamic coefficients, single conductor diameter D, mass per unit length m, moment of inertia I, and frequencies in the vertical, horizontal, and torsional directions. Damping ratio ;

[0016] The dynamics model module is used to establish the motion control matrix equations for transmission lines based on modal displacements. It includes: establishing the dynamic displacement expression of the transmission line structure and expanding it using modal expansion; constructing the expression of the self-excited aerodynamic forces of the icing transmission line based on quasi-steady-state theory; simplifying the aerodynamic expression through a linear force model and obtaining the generalized modal forces through integration; and transforming the dynamic equations into motion control matrix equations that include generalized mass, damping, and stiffness matrices.

[0017] 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.

[0018] 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. By observing the change of the modal damping ratio ξ with the wind speed U, the critical wind speed for transmission line galloping is determined.

[0019] The beneficial effects of this invention are as follows:

[0020] The critical wind speed prediction method for transmission line galloping based on the LP perturbation method disclosed in this invention has the following advantages:

[0021] 1) Improved computational efficiency:

[0022] By simply iterating over the modal frequencies, the modal damping ratio and the corresponding eigenvectors can be obtained directly from the calculated modal frequencies, avoiding the complex iterative calculation process and improving computational efficiency.

[0023] 2) The mechanism is clearly explained:

[0024] Decomposing complex coupled nonlinear problems into a linear superposition of structural, uncoupled, and coupled components helps to clearly reveal the mechanism by which structural and aerodynamic parameters affect aerodynamic instability, and clearly demonstrates the influence of aerodynamic coefficients and structural parameters on aerodynamic instability.

[0025] 3) Improved prediction accuracy:

[0026] It can quickly and accurately obtain the closed-loop solution of the aerodynamic instability initiation condition and clearly reveal the influence of each coupling component on the instability mechanism, thereby improving the accuracy of predicting the critical wind speed for power transmission line galloping.

[0027] 4) Innovation in analytical methods:

[0028] By defining new variables and transforming the differential form, rewriting the governing equations, constructing a perturbation expansion framework, and performing power series expansion of eigenvalue and displacement solutions based on the LP perturbation method, a new analytical method is provided, making the solution process more systematic and accurate.

[0029] 5) High applicability:

[0030] It is applicable to the aerodynamic stability analysis of transmission line structure galloping, especially in strongly coupled environments, and can handle three-degree-of-freedom (3DOF) aerodynamic instability problems, with wide applicability.

[0031] 6) Completeness of the technical solution:

[0032] This invention 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.

[0033] 7) Scientific evidence provided:

[0034] It can provide a scientific basis for the wind-resistant design and operation and maintenance of power transmission lines, help prevent and reduce accidents caused by power transmission line galloping, reduce economic losses, and provide a scientific basis for the wind-resistant design and operation and maintenance of power transmission lines.

[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 critical wind speed prediction method for transmission line galloping based on the LP perturbation method of the present invention;

[0038] Figure 2 For the present invention Figure 1 The schematic diagram of quasi-steady-state aerodynamic forces on the cross-section of the icing conductor in step S2;

[0039] Figure 3 This is a cross-sectional view of a four-split icing conductor in an embodiment of the present invention;

[0040] Figure 4 The aerodynamic coefficient of the four-split icing conductor in this embodiment of the invention;

[0041] Figure 5 This invention presents a comparison of modal characteristics between eigenvalue analysis at different wind speeds and the proposed three-dimensional analytical solution method in Embodiment 1 of the present invention. 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 calculation. Figure 5 (b) Comparison of the modal damping ratio of the vertical, lateral and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculation;

[0042] Figure 6 This refers to the contribution of each component in each modal branch to the total damping under different wind speeds in Embodiment 1 of the present invention. 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 various parameters to the coupled damping at different wind speeds are shown in Embodiment 1 of the present invention; wherein... Figure 7 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling damping in the vertical modal branch; Figure 7 (b) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling damping in the torsional mode branch;

[0044] Figure 8 The contributions of various factors to the coupling frequency under different wind speeds in Embodiment 1 of the present invention are shown; wherein... Figure 8 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling frequency in the vertical modal branch; Figure 8 (b) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling frequency in the torsional mode branch;

[0045] Figure 9 This is a comparison of the modal characteristics of eigenvalue analysis and the proposed three-dimensional analytical solution method in Embodiment 2 of the present invention; wherein Figure 9 (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 calculation. Figure 9 (b) Comparison of the modal damping ratio of the vertical, lateral and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculation;

[0046] Figure 10 This refers to the contribution of each component in the three modal branches to the total damping under different wind speeds in Embodiment 2 of the present invention; wherein... Figure 10 (a) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the vertical modal branch; Figure 10 (b) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the transverse modal branch; Figure 10 (c) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the torsional mode branch;

[0047] Figure 11 This refers to the contribution of each term in the coupling component to the coupling damping under different wind speeds in Embodiment 2 of the present invention; wherein... Figure 11 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling damping in the vertical modal branch; Figure 11 (b) represents the contribution of the vertical coupling term and the torsional coupling term to the coupling damping in the transverse modal branch; Figure 11 (c) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling damping in the torsional mode branch. Detailed Implementation

[0048] 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.

[0049] like Figure 1 The method for predicting the critical wind speed for transmission line galloping based on the LP perturbation method, as shown, includes the following steps:

[0050] S1. Collect basic data of the power transmission line;

[0051] Specifically, the basic data collected for the power transmission lines include: aerodynamic coefficients determined through wind tunnel testing, and the diameter of a single conductor. D Mass per unit length Moment of inertia Vertical frequency Vertical damping ratio Horizontal frequency Lateral damping ratio Frequency of twisting direction Torsional damping ratio .

[0052] S2. Establish the motion control matrix equation for the transmission line based on modal displacement;

[0053] The specific steps are as follows:

[0054] S21. Establish the expression for the dynamic displacement of the structure: the structure 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:

[0055]

[0056] in, , and It is the mode shape; Modal displacement; For the cross-directional position.

[0057] S22. Establish the expression for self-excited aerodynamic forces per unit length: (e.g.) Figure 2 As shown, for transmission lines, the self-excited force per unit length can be expressed as a function of the aerodynamic coefficient, i.e., wind-induced galloping. Consider the galloping of iced transmission lines. Vertical, lateral, and torsional displacements are referred to as in-plane, out-of-plane, and torsional displacements, respectively. The direction of the average wind speed is expressed as an angle. Features.

[0058] Based on quasi-steady-state theory, the self-excited aerodynamic force per unit length is:

[0059]

[0060] in, D The diameter of the wire; , and These are the static lift coefficient, drag coefficient, and pitching moment coefficient obtained through wind tunnel testing, respectively. For vertical self-excited aerodynamic force per unit length; The lateral self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; Relative wind speed; For an effective angle of attack; It is a constant used to control the effect of torsional speed on the effective angle of attack.

[0061] S23. Establish a linear force model: When When this is achieved, the following linear force model can be obtained:

[0062]

[0063] in:

[0064]

[0065] S24. Establish the expression for the generalized modal force: Integrate the self-excited aerodynamic force per unit length to obtain the generalized modal force, as shown in the following expression:

[0066]

[0067] in, For vertical generalized modal forces; It is a transverse generalized modal force; To reverse the generalized modal force;

[0068] S25. Establish the transmission line motion control matrix equation based on modal displacement:

[0069]

[0070] 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.

[0071] Based on this, the control matrix equation can be further expressed as:

[0072]

[0073] 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.

[0074] S3. Derive explicit solutions for the modal frequencies, damping ratios, and modal characteristics of each modal branch based on the LP perturbation method;

[0075] The specific steps are as follows:

[0076] 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:

[0077]

[0078] S32. Rewrite the governing equation: Equation (6a) in step S25 can be replaced with the new independent variable defined in step S31. Rewritten as:

[0079]

[0080] 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.

[0081] Solution vector It should be represented as a linear combination of modal vectors:

[0082]

[0083] in, This represents the modal vector of each branch.

[0084] S33. Constructing the perturbation expansion framework: To ensure the initial linearity of equation (8a) 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:

[0085]

[0086] 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:

[0087]

[0088] S35. Solve for the zeroth-order term: Substitute the expansion from step 34 into the rewritten governing equation (8a) 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:

[0089]

[0090] in, ( ) is a constant related to the initial displacement and velocity.

[0091] S36. Solve the equation for the first-order term: For the first-order term ( ):

[0092]

[0093] Substituting the zeroth-order solution from step S35 into equation (12), 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:

[0094]

[0095] When solving for a particular solution with non-identical eigenvalues, the coefficients obtained from the homogeneous equation terms are:

[0096] S37. Solve the second-order term equation: For the second-order term ( ):

[0097]

[0098] Since the first-order solution obtained in step S36 has a periodic solution, substituting it into equation (14) yields the following equation:

[0099]

[0100] The corresponding eigenvalues ​​and solutions are:

[0101]

[0102] in, The dimensionless coefficients are obtained through the chain rule in the differentiation process.

[0103] 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 (10b), 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.

[0104] For vertical modal branching, and The solutions for its modal frequencies and damping ratios are expressed as follows:

[0105]

[0106] 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 .

[0107] Vertical mode vector As shown below,

[0108]

[0109] 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:

[0110]

[0111] For the transverse modal branch, the solution for the relevant parameters is expressed as follows:

[0112]

[0113] For the torsional modal branch, the solution for the relevant parameters is expressed as follows:

[0114]

[0115] S4. Iteratively solve for the modal frequencies, and then directly solve for the damping ratio, amplitude ratio, and phase difference to determine the critical wind speed for galloping.

[0116] The specific steps are as follows:

[0117] 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.

[0118] 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.

[0119] S43. Determine the critical wind speed for galloping: 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 starting wind speed of galloping can be determined when the damping of a certain mode becomes negative.

[0120] Example 1: Taking the galloping of a three-degree-of-freedom conductor as an example (angle of attack) Vertical gallop at 20 degrees)

[0121] The conductor is a horizontal 244-meter single-span four-split ice-covered conductor (e.g.) Figure 3 (As shown). Diameter of a single conductor. D =0.0285 meters, mass per unit length Moment of inertia The frequencies and damping ratios in the vertical, horizontal, and torsional directions are as follows: Hz, Hz, Hz; The influence coefficient of torsional velocity on effective angle of attack. R Set it to 0. The wind is along the positive outward direction, i.e. The aerodynamic coefficients used are as follows: Figure 4 As shown.

[0122] Figure 5 Demonstrated at the angle of attack The modal frequencies and damping ratios of the vertical, lateral, and torsional modal branches were obtained by eigenvalue analysis and the proposed three-dimensional analytical solution at different wind speeds of 20 degrees. 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 calculation. Figure 5 (b) Comparison of the modal damping ratio of the vertical, lateral and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculation;

[0123] The results show that when the damping ratio is small, the two results are consistent, indicating the accuracy of the analytical solution. However, some differences appear when the absolute value of the damping ratio increases. Similar results are found for amplitude ratio and phase difference, but these are not presented due to space limitations. The analysis results indicate that vertical galloping begins at relatively low wind speeds, while the damping ratios of the lateral and torsional modal branches are positive. The vertical and torsional frequencies increase slightly with increasing wind speed, while the lateral frequency remains almost unchanged. The coupling component contributes significantly to the frequencies of the vertical and torsional modal branches and should not be ignored in the calculation. It should be noted that when using frequencies without coupling components to calculate the damping ratio, the results will deviate from the exact solution, especially at higher wind speeds. Therefore, the influence of the coupling component on the frequency and damping ratio needs to be carefully studied.

[0124] Figure 6 The contribution of each component in each modal branch to the total damping is shown under 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) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the torsional mode branch;

[0125] The results show that in the vertical modal branch, the uncoupled component primarily provides negative damping and causes galloping to occur at lower wind speeds. This uncoupled component corresponds to... This is the well-known Den Hartog criterion. As wind speed increases, the additional negative damping caused by the coupled components increases significantly. This is especially true when the vertical and torsional mode frequencies approach each other with increasing wind speed (e.g., ...). Figure 5 (as shown in (a)), the contribution of the coupling term gradually becomes more pronounced. In the torsional modal branch, due to the assumption... R =0, and the aerodynamic damping comes only from the coupling component.

[0126] Figure 7 The contributions of various factors to coupled damping at different wind speeds are shown. Figure 7 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling damping in the vertical modal branch; Figure 7 (b) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling damping in the torsional mode branch;

[0127] Clearly, only a finite number of terms need to be considered in the analysis. In the vertical and torsional modal branches, term 3 in the torsional coupling component and term 2 in the vertical coupling component both correspond to... The reverse effect of the coupling term on different modal branches is also observed here. Therefore, the influence of the coupling components can be easily predicted and explained using the proposed analytical solution.

[0128] Figure 8 The contribution of various factors to the coupling frequency under different wind speeds is shown. Figure 8 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling frequency in the vertical modal branch; Figure 8 (b) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling frequency in the torsional mode branch;

[0129] from Figure 8 It can be clearly seen that term 3 from the torsional coupled motion in the vertical modal branch, and term 4 from the vertical coupled motion in the torsional modal branch, play important roles. It is conceivable that if the relationship with... The related terms will cause a significant deviation between the predicted frequency and the actual frequency, and further cause differences in the damping ratio.

[0130] Example 2: Taking the galloping of a three-degree-of-freedom conductor as an example (angle of attack) Torsional gallop at 45 degrees)

[0131] This embodiment uses the same 244-meter horizontal single-span four-split icing conductor as in Embodiment 1.

[0132] Figure 9 Demonstrated at the angle of attack At 45 degrees, the modal frequencies and damping ratios predicted by the proposed method are consistent with the eigenvalue analysis results, demonstrating the accuracy of the analytical solution. The modal frequencies and damping ratios of the vertical, lateral, and torsional modal branches are calculated using eigenvalue analysis and the proposed three-dimensional analytical solution. Figure 9 (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 calculation. Figure 9 (b) Comparison of the modal damping ratio of the vertical, lateral and torsional modal branches with wind speed under eigenvalue analysis and three-dimensional analytical solution calculation;

[0133] The results show that galloping was observed only in the torsional mode branch at a wind speed of 5.3 m / s. Furthermore, the results were compared with those obtained using analytical solutions that do not consider the influence of coupled frequency components. Although the torsional frequency decreases with increasing wind speed, the frequency intervals are relatively large within the calculated wind speed range, so the coupled frequency components can be ignored in frequency prediction. Moreover, as... Figure 9 As shown in (b), the predicted damping value differs slightly from the accurate result only at high wind speeds. It can be anticipated that as wind speed increases, the frequency interval between the torsional and lateral modal branches decreases, and the coupling interaction effect between the lateral and torsional modal branches will be enhanced.

[0134] Figure 10 The contribution of each component in the three modal branches to the total damping is shown at different wind speeds. Figure 10 (a) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the vertical modal branch; Figure 10 (b) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the transverse modal branch; Figure 10 (c) represents the contributions of structural damping, uncoupled damping, and coupled damping to the total damping in the torsional mode branch;

[0135] The results show that in the torsional mode branch, the coupling component provides negative damping for the system and induces galloping.

[0136] Figure 11 The contributions of each term in the coupling component to the coupling damping at different wind speeds are given. Figure 11 (a) represents the contribution of the transverse coupling term and the torsional coupling term to the coupling damping in the vertical modal branch; Figure 11 (b) represents the contribution of the vertical coupling term and the torsional coupling term to the coupling damping in the transverse modal branch; Figure 11(c) represents the contribution of the vertical coupling term and the transverse coupling term to the coupling damping in the torsional modal branch;

[0137] Clearly, both lateral and vertical coupling provide negative damping. In particular, term 2 in the vertical coupling component (i.e....) ) and term 2 in the transverse coupling component (i.e. This is the dominant term. Under this condition, the importance of lateral coupled motion increases with wind speed. While the above term is equally critical due to the inverse effect of coupled motion, it provides additional positive damping for the modal branch (e.g., ...). Figure 10-11 (As shown). Due to the large frequency interval, the coupling effect of vertical-lateral motion can be ignored. It is worth noting that although lateral coupling motion induces negative damping in other modal branches, galloping does not occur in the lateral modal branches. This is because... The aerodynamic positive damping is relatively large. Therefore, three-degree-of-freedom analysis is particularly important when studying the galloping mechanism of transmission lines.

[0138] 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 wind speed for transmission line galloping based on the LP perturbation method, characterized in that, Includes the following steps: S1. Obtain the physical properties and aerodynamic parameters of the transmission line as basic data through wind tunnel testing; S2. Establish the motion control matrix equation for the transmission line based on modal displacement, which includes: establishing the dynamic displacement expression of the transmission line structure and expanding it using modal expansion; constructing the expression of the self-excited aerodynamic force of the icing transmission line based on quasi-steady-state theory; simplifying the aerodynamic force expression through a linear force model and obtaining the generalized modal force through integration; and transforming the dynamic equation into a motion control matrix equation containing the generalized mass, damping, and stiffness matrices. 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; Specifically: S31. Define a new variable and transform the differential form: Define a new independent variable. And perform partial derivatives on the eigenvalues ​​of each modal branch; S32. Rewrite the governing equations: Using the equations from step S31 The established transmission line motion control matrix equations based on modal displacement are rewritten as follows: 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; , These are the generalized mass matrix and the damping matrix, respectively. It is a generalized displacement vector; and These are the aerodynamic stiffness matrix and the aerodynamic damping matrix; S33. Construct a perturbation expansion framework: Represent the mass matrix, damping matrix, and stiffness matrix as the sum of a zero-order matrix and a first-order matrix; S34. Based on the LP perturbation method, perform a power series expansion of the eigenvalues ​​and displacement solutions: eigenvalues and displacement solution according to The power series expansion is: S35. Solve for the zeroth-order term: Substitute formulas (10a) and (10b) from step 34 into formula (8a) in 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... Displacement solution For the corresponding 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... : Substitute the zero-order term in step S35 into equation (12), and then set the coefficients of each long-term term to zero to obtain the corresponding first-order eigenvalues ​​and first-order solutions. S37. Solve the second-order term equation: For the second-order term... : And solve for the corresponding second-order eigenvalues ​​and second-order solutions; 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 (10b), through The solution is periodic, and the modal frequencies of each modal branch are... Damping ratio Through eigenvalues Solve for, where That is, to obtain explicit solutions for the frequency and damping ratio of each modal branch; S4. Iteratively solve for the modal frequencies of each modal branch. Based on this, directly solve for the damping ratio, amplitude ratio, and phase difference, and observe the modal damping ratio in each direction. With wind speed The changes were used to determine the critical wind speed for power line galloping.

2. The method for predicting the critical wind speed for power transmission line galloping as described in claim 1, characterized in that, The basic data of the transmission line collected in step S1 includes: aerodynamic coefficients determined by wind tunnel tests, and the diameter of a single conductor. Mass per unit length Moment of inertia Vertical frequency Vertical damping ratio Horizontal frequency Lateral damping ratio Frequency of twisting direction Torsional damping ratio .

3. The method for predicting the critical wind speed for power transmission line galloping as described in claim 2, characterized in that, Step S2 is as follows: S21. Establish the expression for the dynamic displacement of the structure: the structure 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: （1） in, It is the mode shape; For modal displacement; For the cross-directional position; S22. Establish the expression for the self-excited aerodynamic force per unit length: Based on quasi-steady-state theory, the expression for the self-excited aerodynamic force per unit length considering the galloping of iced transmission lines is as follows: The direction of the average wind speed is expressed as an angle. As a characteristic; The diameter of the wire; These are the static lift coefficient, drag coefficient, and pitching moment coefficient obtained through wind tunnel testing, respectively. For vertical self-excited aerodynamic force per unit length; The lateral self-excited aerodynamic force per unit length; To generate self-excited aerodynamic force per unit length for torsion; Relative wind speed; For an effective angle of attack; This is a constant used to control the effect of torsional speed on the effective angle of attack; S23. Establish a linear force model: When At that time, the simplified linear force model expression is: in: S24. Establish the expression for the generalized modal force: The generalized modal force is obtained by integrating the self-excited aerodynamic force per unit length, and the expression is as follows: in, For vertical generalized modal forces; It is a transverse generalized modal force; To reverse the generalized modal force; S25. Establish the transmission line motion control matrix equation based on modal displacement: 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.

4. The method for predicting the critical wind speed for power transmission line galloping as described in claim 3, characterized in that, The control matrix equation (5a) in step S25 is further expressed as: in, These are the uncoupled damping ratio and frequency caused by the uncoupled self-excited force, respectively. Let be the dimensionless mass and the moment of inertia.

5. The method for predicting the critical wind speed for power transmission line galloping as described in claim 4, characterized in that, Step S31 specifically involves: applying the differential chain rule to convert the time derivative into partial derivatives with respect to the eigenvalues ​​of each modal branch. Equation (6a) in step S25 uses the new independent variable defined in step S31. rewrite; Solving for the generalized displacement vector It should be represented as a linear combination of modal vectors: in, Represents the modal vectors of each branch; Step S33 specifically involves: to ensure the initial linearity of equation (8a) in step S32 and the feasibility of each order of iteration, the mass matrix, damping matrix, and stiffness matrix are expressed as the sum of a zero-order matrix and a first-order matrix, where... The small perturbation parameter of the nonlinear term in the equation is, i.e. The specific expression is: The first-order eigenvalues ​​and first-order solutions in step S36 are: When solving for a particular solution with non-identical eigenvalues, the coefficients obtained from the homogeneous equation terms are: ; Since the first-order solution obtained in step S36 has a periodic solution, substituting it into equation (14) yields the following equation: The second-order eigenvalues ​​and second-order solutions obtained in step S37 are: in, The dimensionless coefficients are obtained through the chain rule in the differentiation process.

6. The method for predicting the critical wind speed for power transmission line galloping as described in claim 5, 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 Let be the similarity factor between each mode shape; in particular, the frequency of the coupled modal branch is 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 and 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: 。 7. The method for predicting the critical wind speed for power transmission line galloping as described in claim 6, characterized in that, Step S4 specifically involves iteratively solving for the modal frequencies of each modal 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 The damping ratio is gradually increased until it becomes negative, at which point the critical wind speed for power line galloping is determined.

8. A transmission line galloping critical wind speed prediction system based on the LP perturbation method, used to implement the transmission line galloping critical wind speed prediction method according to any one of claims 1 to 7, characterized in that, include: The data acquisition module is used to obtain the physical characteristics and aerodynamic parameters of the transmission line, including aerodynamic coefficients and the diameter of a single conductor. Mass per unit length Moment of inertia and frequencies in the vertical, horizontal and torsional directions. Damping ratio ; The dynamics model module is used to establish the motion control matrix equations for transmission lines based on modal displacements. It includes: establishing the dynamic displacement expression of the transmission line structure and expanding it using modal expansion; constructing the expression of the self-excited aerodynamic forces of the icing transmission line based on quasi-steady-state theory; simplifying the aerodynamic expression through a linear force model and obtaining the generalized modal forces through integration; and transforming the dynamic equations into motion control matrix equations that include 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 modal frequencies and damping ratios of each modal branch; The wind speed prediction module iteratively solves for the modal frequencies of each modal branch, and directly calculates the damping ratio, amplitude ratio, and phase difference based on these results. The modal damping ratio is observed during the calculation. With wind speed The changes in wind speed were used to determine the critical wind speed for power line galloping.

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