A method for calculating the propagation velocity of high-speed railway contact wire fluctuations based on an analytical model

Abstract

This invention discloses a method for calculating the wave propagation velocity of high-speed railway contact wire based on an analytical model. Specifically, the method involves: first, determining the differential equation of the vertical motion of the contact wire, and then solving the differential equation using the residue theorem; analyzing the exponential terms of the solution to determine the wavelength of the wave on the contact wire, and obtaining the wave propagation velocity based on the harmonic frequencies and their corresponding wavelengths; then, establishing an analytical model of an infinitely long viscoelastic layer, solving the analytical model, and obtaining the vertical displacement of the contact wire; adjusting the contact wire damping value to match the attenuation coefficient of the analytical model to obtain the contact wire damping value, and using this damping to solve for the wave propagation velocity; finally, adjusting the harmonic frequencies, calculating the contact wire damping values ​​at different frequencies, determining the dominant vibration frequency of the pantograph-catenary coupling, and solving for the wave propagation velocity at that frequency. This invention offers higher computational efficiency and improves the accuracy of the contact wire wave propagation velocity calculation.

Description

Technical Field

[0001] This invention belongs to the field of catenary wave behavior simulation technology, and particularly relates to a method for calculating the wave propagation speed of high-speed railway catenary based on an analytical model. Background Technology

[0002] The overhead contact line, commonly known as the catenary, is a system that provides electrical energy to high-speed railway vehicles through sliding contact with the pantograph. Good pantograph-catenary dynamic performance is crucial for the safe operation of high-speed trains. The undulation performance of the contact wire is a key research area in high-speed pantograph-catenary systems, and the undulation speed is the main indicator for measuring this performance.

[0003] Currently, the widely accepted formula for calculating wave propagation velocity comes from the European standard EN50119, which is based on mathematical derivation, field experiments, and engineering experience. Zou Dong et al. used a non-contact photogrammetry device to measure and verify the wave group velocity of the contact wire, showing a discrepancy between the wave velocity obtained by the traditional calculation method and the experimental results. Contact wire damping, as a crucial factor in contact network dynamics, has not been considered in most studies of wave propagation velocity calculation. Liu Zhigang et al. considered the influence of air damping and corrected the calculation method for contact wire wave propagation velocity; since the air damping caused by static wind load is relatively small, it does not significantly affect the calculation of contact wire wave propagation velocity. Park et al. analyzed the test signals of contact network vibration, explored the dispersion problem of waves on the contact network, and presented the relationship between wave propagation velocity and wave frequency on the contact network. Li Fuzhong et al. used numerical analysis methods to analyze the relationship between wavelength and wave propagation velocity and proposed methods to improve the utilization rate of contact network wave velocity.

[0004] In summary, existing research on the propagation speed of contact wire ripples in high-speed railways largely focuses on analytical model calculations, numerical simulations, and track tests. However, traditional methods for calculating the propagation speed of contact wire ripples cannot achieve accurate calculations, and finite element simulation methods suffer from low computational efficiency. Physical tests also face limitations imposed by on-site conditions and high testing costs. Therefore, a new method for accurately and efficiently calculating the propagation speed of contact wire ripples is urgently needed. Summary of the Invention

[0005] To address the above problems, this invention provides a method for calculating the propagation speed of high-speed railway contact wire fluctuations based on an analytical model.

[0006] This invention discloses a method for calculating the wave propagation velocity of high-speed railway contact wire based on an analytical model. Based on the wave attenuation law and the analytical model of the contact network, the method obtains the damping of the contact wire, considers the influence of the contact wire damping, and applies the wave propagation law and residue theorem to accurately calculate the wave propagation velocity of the contact wire. Specifically, it includes the following steps:

[0007] Step 1: Determine the differential equation of vertical motion of the contact line, as shown in equation (1):

[0008]

[0009] Where: μ c and c m Let represent the mass and damping per unit length, respectively; T represent the contact wire tension; F0 represent the contact force; δ represent the Dirac function; w represent the vertical displacement of the contact wire; x represent the longitudinal position of the contact wire; Ω represent the frequency; v represent the pantograph's operating speed; t represent time; and i represent the imaginary unit.

[0010] Step 2: Using the residue theorem, solve the differential equation of the contact line. The result is shown in equation (2):

[0011]

[0012] in:

[0013]

[0014] μ represents the mass per unit length, and the superscripts and subscripts c represent the contact wire.

[0015] in:

[0016]

[0017] Step 3: Assuming the pantograph's operating speed is 0, the solution for the contact wire is obtained, as shown in equation (5):

[0018]

[0019] Step 4: Analyze the exponential term of equation (5) to determine the wavelength of the wave on the contact line:

[0020]

[0021] in: represent The real part.

[0022] Based on the harmonic frequency and its corresponding wavelength, the wave propagation speed of the contact wire is obtained as follows:

[0023]

[0024] Step 5: Establish an analytical model of the infinitely long viscoelastic layer, solve the analytical model, and obtain the expression for the vertical displacement of the contact line:

[0025]

[0026] p and q represent positive and negative numbers respectively, and r refers to the pole in the formula whose sign is not the same as p or q. Therefore, k p k represents the pole where the imaginary part is positive. q This represents the pole where the imaginary part is negative.

[0027] in:

[0028]

[0029] in:

[0030]

[0031] in:

[0032]

[0033] α and β refer to the proportional damping coefficients. The average stiffness of the analytical model.

[0034] Step 6: Determine the proportional attenuation coefficient in the contact wire model. The wave will be formed along the positive x-direction. proportional decay, yes The imaginary part; similarly, in the analytical model of an infinitely long viscoelastic layer, k 1I k is the proportional attenuation coefficient. 1I It is the imaginary part of k1.

[0035] Step 7: Adjust the contact wire damping value c m The magnitude is determined so that the attenuation coefficient of the analytical model matches the attenuation coefficient of the contact wire, the damping value of the contact wire is obtained, and the wave propagation speed is solved using the damping.

[0036] Step 8: Adjust the harmonic frequency, repeat step 7, and calculate the damping value of the contact wire at different frequencies.

[0037] Step 9: Determine the dominant frequency of the pantograph-catenary coupling, and based on the dominant frequency of the pantograph-catenary coupling, solve for the contact wire ripple propagation speed at that frequency.

[0038] The beneficial technical effects of this invention are as follows:

[0039] The proposed method for calculating the wave propagation velocity of high-speed railway contact wires utilizes an analytical model of the contact network to calculate this velocity. This avoids the limitations of on-site conditions and high costs associated with physical testing, and offers higher computational efficiency compared to finite element simulation. Compared to previous studies that neglected or roughly estimated damping, this invention combines an analytical model with wave attenuation theory to accurately solve for the contact wire damping at different frequencies. This method significantly improves the accuracy of contact wire wave propagation velocity calculations compared to traditional methods. Attached Figure Description

[0040] Figure 1 This is a flowchart illustrating the method for calculating the propagation speed of high-speed rail contact wire fluctuations based on an analytical model, as presented in this invention.

[0041] Figure 2 The results of the contact line wave velocity calculation using the traditional wave propagation velocity calculation method and the wave propagation velocity calculation method proposed in this invention are ((a) is the Beijing-Tianjin Railway; (b) is EN50318:2018).

[0042] Figure 3 These are the damping values ​​of the contact wire at different frequencies. Method 1 obtains the damping value by analyzing wave attenuation, while Method 2 obtains the damping value using the proportional damping formula.

[0043] Figure 4 The dominant frequency of the pantograph-catenary coupling is ((a) for the Beijing-Tianjin Railway; (b) for EN50318:2018).

[0044] Figure 5 These are contour maps of contact line displacement ((a) is the Beijing-Tianjin Railway; (b) is EN50318:2018). Detailed Implementation

[0045] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0046] The implementation example uses the line data of the Beijing-Tianjin Railway and the standard data of EN50318:2018 to verify the proposed method for calculating the propagation speed of high-speed rail contact wire fluctuations based on the analytical model.

[0047] The flowchart of the method for calculating the propagation velocity of high-speed rail contact wire fluctuations based on an analytical model is as follows: Figure 1 As shown, based on the wave attenuation law and the analytical model of the contact network, the damping of the contact wire is obtained. Considering the influence of the contact wire damping, the wave propagation law and residue theorem are applied to accurately calculate the wave propagation speed of the contact wire. Specifically, the following steps are included:

[0048] Step 1: Based on nonlinear finite element theory, a three-dimensional model of the contact network is established using nonlinear cable and rod elements. Combined with a three-dimensional mass pantograph model, dynamic simulation of the pantograph-contact network is achieved. The effectiveness of the dynamic solution is verified using data from EN50318:2018.

[0049] Step 2: Determine the differential equation of vertical motion of the contact line, as shown in equation (1):

[0050]

[0051] Where: μ c and c m Let represent the mass and damping per unit length, respectively; T represent the contact wire tension; F0 represent the contact force; δ represent the Dirac function; w represent the vertical displacement of the contact wire; x represent the longitudinal position of the contact wire; Ω represent the frequency; v represent the pantograph's operating speed; t represent time; and i represent the imaginary unit.

[0052] Step 3: Using the residue theorem, solve the differential equation of the contact line. The result is shown in equation (2):

[0053]

[0054] in:

[0055]

[0056] μ represents the mass per unit length, and the superscripts and subscripts c represent the contact wire.

[0057] in:

[0058]

[0059] Step 4: Assuming the pantograph's operating speed is 0, the solution for the contact wire is obtained, as shown in equation (5):

[0060]

[0061] Step 5: Analyze the exponential term of equation (5) to determine the wavelength of the wave on the contact line:

[0062]

[0063] in: represent The real part.

[0064] Based on the harmonic frequency and its corresponding wavelength, the wave propagation speed of the contact wire is obtained as follows:

[0065]

[0066] The contact line wave propagation velocity obtained by the traditional method and the method proposed in this invention is as follows: Figure 2 As shown.

[0067] Step 6: Establish an analytical model of the infinitely long viscoelastic layer, solve the analytical model, and obtain the expression for the vertical displacement of the contact line:

[0068]

[0069] p and q represent positive and negative numbers respectively, r refers to the pole in the formula whose sign is not the same as p or q, and k p k represents the pole where the imaginary part is positive. q This represents the pole where the imaginary part is negative.

[0070] in:

[0071]

[0072] in:

[0073]

[0074] in:

[0075]

[0076] α and β refer to the proportional damping coefficients. The average stiffness of the analytical model.

[0077] Step 7: Determine the proportional attenuation coefficient in the contact wire model. The wave will be formed along the positive x-direction. proportional decay, yes The imaginary part; similarly, in the analytical model of an infinitely long viscoelastic layer, k 1I k is the proportional attenuation coefficient. 1I It is the imaginary part of k1.

[0078] Step 8: Adjust the contact wire damping value c m The magnitude is determined so that the attenuation coefficient of the analytical model matches the attenuation coefficient of the contact wire, the damping value of the contact wire is obtained, and the wave propagation speed is solved using the damping.

[0079] Step 9: Adjust the harmonic frequency, repeat step 8, and calculate the damping value of the contact wire at different frequencies, such as... Figure 3 As shown, when the system frequency is low, the damping value obtained by method 1 is significantly greater than that obtained by method 2.

[0080] Step 10: Using the model established in Step 1, obtain the principal frequencies of the pantograph-catenary coupling, such as... Figure 4As shown, the main oscillation frequency of the Beijing-Tianjin Railway is 1.3Hz, while the main oscillation frequency of EN50318:2018 is 0.9Hz.

[0081] Step 11: Using the obtained pantograph-catenary coupling master frequency, calculate the contact wire ripple propagation velocity at that frequency, such as... Figure 2 As shown.

[0082] Step 12: Using the pantograph-catenary coupling model established in Step 1, obtain the contact line displacement contour map as follows: Figure 5 As shown in the contour map, the slope is determined as the wave propagation speed, and the average wave propagation speed of forward and backward propagating waves is taken as the measured value of wave propagation speed. The calculated and measured values ​​of wave propagation speed are shown in Table 1.

[0083] Table 1. Calculated and measured values ​​of wave propagation speed in the examples.

[0084]

[0085] The method proposed in this invention has a maximum relative error of 2.71%, while the traditional calculation method has a minimum relative error of 7.05% and a maximum of 12.18%. In summary, compared with the traditional calculation method, the proposed method for calculating the wave propagation speed of high-speed rail contact wire based on the analytical model has a smaller relative error and obtains a more accurate wave propagation speed.

Claims

1. A method for calculating the propagation velocity of high-speed railway contact wire fluctuations based on an analytical model, characterized in that, Includes the following steps: Step 1: Determine the differential equation of vertical motion of the contact line, as shown in equation (1): Where: μ c and c m Let represent the mass and damping per unit length, respectively; T represents the contact wire tension; F0 represents the contact force; δ represents the Dirac function; w represents the vertical displacement of the contact wire; x represents the longitudinal position of the contact wire; Ω represents the frequency; v represents the pantograph's operating speed; t represents time; and i represents the imaginary unit. Step 2: Using the residue theorem, solve the differential equation of the contact line. The result is shown in equation (2): in: μ represents the mass per unit length, and the superscript and subscript c represent the contact wire. in: Step 3: Assuming the pantograph's operating speed is 0, the solution for the contact wire is obtained, as shown in equation (5): Step 4: Analyze the exponential term of equation (5) to determine the wavelength of the wave on the contact line: in: represent The real part; Based on the harmonic frequency and its corresponding wavelength, the wave propagation speed of the contact wire is obtained as follows: Step 5: Establish an analytical model of the infinitely long viscoelastic layer, solve the analytical model, and obtain the expression for the vertical displacement of the contact line: p and q represent positive and negative numbers respectively, r refers to the pole in the formula whose sign is not the same as p or q, and k p k represents the pole where the imaginary part is positive. q This represents the poles where the imaginary part is negative; in: in: in: α and β refer to the proportional damping coefficients. The average stiffness of the analytical model; Step 6: Determine the proportional attenuation coefficient in the contact wire model. The wave will be formed along the positive x-direction. proportional decay, yes The imaginary part; similarly, in the analytical model of an infinitely long viscoelastic layer, k 1I k is the proportional attenuation coefficient. 1I It is the imaginary part of k1; Step 7: Adjust the contact wire damping value c m The magnitude is determined so that the attenuation coefficient of the analytical model matches the attenuation coefficient of the contact wire, the damping value of the contact wire is obtained, and the wave propagation speed is solved using the damping. Step 8: Adjust the harmonic frequency, repeat step 7, and calculate the damping value of the contact wire at different frequencies; Step 9: Determine the dominant frequency of the pantograph-catenary coupling, and based on the dominant frequency of the pantograph-catenary coupling, solve for the contact wire ripple propagation speed at that frequency.

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