Optimization method for linear equation of cubic parabola transition curve in railway and rail transportation

Abstract

The optimization method of the linear equation of the cubic parabola transition curve of railways and rail transit is to ensure that the actual calculated length of the transition curve is consistent with the design required length, the curvature radius of the transition point or the calculated curvature radius of the circular transition point is consistent with the design required radius, and the smooth connection between the circular curve and the transition curve is ensured, so as to ensure the safety of construction and operation of railways and other rail transit and improve riding comfort. It includes the following steps: Step 1, first meet the design required length of the transition curve and calculate the parameter X value; Step 2, meet the design required radius and calculate the parameter J. The equation of the cubic parabola transition curve after constant optimization, the curvature radius of any point on the transition curve and the length from the straight transition point or the transition point are calculated as follows:

Description

Technical Field

[0001] The present invention relates to the field of civil engineering, and in particular to an optimization method for a linear equation of a cubic parabola transition curve for railways and rail transportation. Background Art

[0002] Railways and wheel-rail rail transit should use cubic parabolas as transition curves, smoothly connecting the circular curves and straight lines to form the route design line. However, during design and surveying, calculations based on the current equation for the cubic parabola transition curve revealed that the actual calculated length of the transition curve and the design required length, as well as the calculated curvature radius of the transition point or the circular transition point and the design required radius, could not be simultaneously the same. This resulted in the circular curve and the transition curve not being able to be smoothly closed, which would affect the safety and smoothness of line operation and make it unusable for design, construction, and maintenance. Research on methods to optimize the constants of the current cubic parabola transition curve equation and propose a correct cubic parabola transition curve equation is extremely important for line design, construction, maintenance, and software development for railways and various types of rail transit that use cubic parabola transition curves.

[0003] During design, the calculated length of the transition curve and the calculated curvature radius of the transition point or circular transition point are the actual design values. When a cubic parabola is used as a transition curve, and the current equation is used for calculation and design, if the transition curve length is consistent with the design requirement, the actual calculated curvature radius of the transition curve transition point or circular transition point will differ from the design requirement. Conversely, if the calculated curvature radius of the transition curve transition point or circular transition point is forced to be the same as the design requirement, the actual calculated length of the transition curve will not be consistent with the design requirement. In other words, the actual calculated curvature radius of the transition point or circular transition point and the transition curve length cannot simultaneously meet the design requirement parameters. This means that the difference between the calculated value using the original cubic parabola transition curve equation and the design requirement is significant, resulting in the inability of the cubic parabola transition curve and the circular curve to close smoothly according to the design requirements, significantly impacting line safety and smoothness. In order to ensure that the length and geometric shape of the cubic parabola transition curve in design results, surveying, construction and maintenance meet the design requirements and improve the safety and smoothness of line operation, the constant C value of the original cubic parabola plane equation must be optimized. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide an optimization method for the linear equation of the cubic parabola transition curve of railways and rail transit, so as to ensure that the actual calculated length of the transition curve is consistent with the design required length, the transition point or the calculated curvature radius of the circular transition point is consistent with the design required radius, and the smooth connection between the circular curve and the transition curve is ensured, so as to ensure the safety of construction and operation of railways and other rail transit, and improve riding comfort. The method can be used in engineering for railway and rail transit line design, surveying, construction, overhaul, software development and theoretical research.

[0005] The technical solutions adopted by the present invention to solve the technical problems are as follows:

[0006] The method for optimizing the linear equation of a cubic parabola transition curve for railways and rail transit according to the present invention comprises the following steps:

[0007] Step 1: First, meet the required length of the transition curve and calculate the parameter X value

[0008] ① When the straight-line transition point is taken as the origin and the straight-line side is taken as the x-axis, the linear equation of the cubic parabola transition curve is as follows (1):

[0009]

[0010] Where: C is a constant, which is the product of the required circular curve radius R and the required length l0 of the transition curve; x and y are the plane coordinates of any point on the transition curve; h is the superelevation value of any point on the transition curve; and h0 is the superelevation design value of the curve.

[0011] In order to make the actual calculated length of the cubic parabola transition curve consistent with the design required length, let X be the x-coordinate value of the transition point of the cubic parabola transition curve, and replace l0 in the first line of equation (1) with the parameter X. Then the plane equation of the cubic parabola transition curve evolves as follows:

[0012]

[0013] ② Using calculus analysis, let the length of the small segment on the transition curve be dl. According to formula (1), the first-order derivatives of the length of the small segment on the transition curve and the length of the transition curve with respect to x are as follows:

[0014]

[0015]

[0016] ③ Using the original cubic parabola equation, the formula for calculating the radius of curvature of any point on the transition curve is derived through formula (1). The formula for calculating the distance from any point on the transition curve to the straight-slow point or the slow-straight point is derived through formula (4) and by setting C = RX. They are as follows:

[0017]

[0018] Where: ρ and l are the curvature radius of any point on the transition curve and the distance from the straight-to-slow point or the slow-to-straight point on the transition curve;

[0019] ④ Calculate the X value. Let l = l0 and x = X in formula (5). Substitute the required length l0 and required radius R of the transition curve in the design specification into the second line of formula (5) to calculate the X value of each curve.

[0020] Step 2: Meet the design requirement radius and calculate parameter J

[0021] ① In order to make the calculated curvature radius of the transition point or the transition point of the cubic parabola transition curve the same as the design requirement radius, replace R in formula (1) with the parameter J, and the cubic parabola transition curve equation evolves as follows:

[0022]

[0023] Where: J is the optimization parameter of the cubic parabola transition curve equation, which is close to the design radius;

[0024] ② From formula (6), the calculation formula for the curvature radius of any point on the transition curve can be evolved into:

[0025]

[0026] ③ According to formula (7), with the straight-slow point as the origin of the rectangular coordinate system, the curvature radius of the slow-circle point or the slow-circle point is calculated as:

[0027]

[0028] Where: R' is the calculated value of the curvature radius of the transition point or the transition point of the cubic parabola transition curve;

[0029] ④ Calculate the J value. Substitute the required length of the transition curve and the X value calculated in step 1 into formula (8), and let R' = R. The parameter J value of each curve can be calculated. Substitute the J value and the required length of the transition curve into formula (6), and let the constant C' = Jl0. The equation of the cubic parabola transition curve after constant optimization, the curvature radius of any point on the transition curve, and the length from the straight-slow point or the slow-straight point are obtained as follows:

[0030]

[0031] Where: J is the optimization parameter of the cubic parabola transition curve equation, which is calculated using formula (8); C' is a constant, which is the product of parameter J and the required length l0 of the transition curve during the optimization process; x and y are the plane coordinates of any point on the transition curve; ρ and l are the curvature radius of any point on the transition curve and the distance from the straight-slow point or the slow-straight point on the transition curve.

[0032] The beneficial effect of the present invention is that the constants and linear equations of the original cubic parabola transition curve equation are optimized by a calculation method. By using the optimized equation (9), the actual calculated length of the transition curve and the design required length, the transition point or the calculated curvature radius of the circular transition point and the design required radius can be simultaneously the same, and the circular curve and the transition curve are smoothly connected. This provides a technical basis for the line design, surveying, maintenance, theoretical research and compilation of design software for transportation projects such as railways, rail transportation and roads that use cubic parabolas as transition curves. The invention is widely used and has good practicality and versatility. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a schematic diagram of a cubic parabola transition curve in a rectangular coordinate system. DETAILED DESCRIPTION

[0034] The present invention will be further described below with reference to the accompanying drawings and examples.

[0035] The comparison between the actual calculated curvature radius of the transition point or circular transition point of the cubic parabola transition curve and the actual calculated length of the transition curve and the design requirements is shown in the following table.

[0036] Through calculation based on formula (5) and the design requirement values ​​of the current specifications, it is found that when the original cubic parabola equation is used, if the actual calculated length of the transition curve is consistent with the design requirement length, the actual calculated curvature radius of the transition curve transition point or the circular transition point is different from the design requirement radius. On the contrary, if the calculated curvature radius of the transition curve transition point or the circular transition point is the same as the design requirement radius, the actual calculated length of the transition curve is inconsistent with the design requirement length, that is, the actual calculated curvature radius of the transition point or the circular transition point and the transition curve length cannot simultaneously meet the design requirement radius and the design requirement transition curve length.

[0037] Table 1 Comparison of actual calculated curvature radius, transition curve length and design requirements

[0038]

[0039] Table 1 shows that the values ​​calculated using the original cubic parabola transition curve equation differ significantly from the design requirements. This prevents the cubic parabola transition curve and the circular curve from closing smoothly according to the design parameters, significantly impacting line safety and smoothness. To ensure that the length and geometric shape of the cubic parabola transition curve meet the requirements for design, installation, construction, and maintenance, and to improve line safety and smoothness, the constant C in the original cubic parabola plane equation must be optimized.

[0040] Based on this, the constants of the cubic parabola transition curve plane equation in formula (1) must be optimized.

[0041] Reference Figure 1The method for optimizing the linear equation of a cubic parabola transition curve for railways and rail transit of the present invention comprises the following steps:

[0042] Step 1: First, meet the required length of the transition curve and calculate the parameter X value

[0043] ① When the straight-line transition point is taken as the origin and the straight-line side is taken as the x-axis, the linear equation of the cubic parabola transition curve is as follows (1):

[0044]

[0045] The linear equation of the cubic parabola transition curve comes from the results of the scientific and technological research project "Research on the Selection of Design Parameters of Bridges and Tunnels for High-Speed ​​Railway Lines" and is a requirement of the current railway and various types of rail transit design specifications.

[0046] Where: C is a constant, which is the product of the required circular curve radius R and the required length l0 of the transition curve; x and y are the plane coordinates of any point on the transition curve; h is the superelevation value of any point on the transition curve; and h0 is the superelevation design value of the curve.

[0047] In order to make the actual calculated length of the cubic parabola transition curve consistent with the design required length, let X be the x-coordinate value of the transition point of the cubic parabola transition curve, and replace l0 in the first line of equation (1) with the parameter X. Then the plane equation of the cubic parabola transition curve evolves as follows:

[0048]

[0049] ② Using calculus analysis, let the length of the small segment on the transition curve be dl. According to formula (1), the first-order derivatives of the length of the small segment on the transition curve and the length of the transition curve with respect to x are as follows:

[0050]

[0051] ③ Using the original cubic parabola equation, the formula for calculating the radius of curvature of any point on the transition curve is derived through formula (1). The formula for calculating the distance from any point on the transition curve to the straight-slow point or the slow-straight point is derived through formula (4) and by setting C = RX. They are as follows:

[0052]

[0053] Where: ρ and l are the curvature radius of any point on the transition curve and the distance from the straight-to-slow point or the slow-to-straight point on the transition curve;

[0054] ④ Calculate the X value. Let l = l0 and x = X in formula (5). Substitute the required length l0 and required radius R of the transition curve in the design specification into the second line of formula (5) to calculate the X value of each curve.

[0055] Step 2: Meet the design requirement radius and calculate parameter J

[0056] ① In order to make the calculated curvature radius of the transition point or the transition point of the cubic parabola transition curve the same as the design requirement radius, replace R in formula (1) with the parameter J, and the cubic parabola transition curve equation evolves as follows:

[0057]

[0058] Where: J is the optimization parameter of the cubic parabola transition curve equation, which is close to the design radius;

[0059] ② From formula (6), the calculation formula for the curvature radius of any point on the transition curve can be evolved into:

[0060]

[0061] ③ According to formula (7), with the straight slow point as the origin, the curvature radius of the slow circular point or the circular slow point is calculated as:

[0062]

[0063] Where: R' is the calculated value of the curvature radius of the transition point or the transition point of the cubic parabola transition curve;

[0064] ④ Calculate the J value. Substitute the required length of the transition curve and the X value calculated in step 1 into formula (8), and let R' = R. The parameter J value of each curve can be calculated. Substitute the J value and the required length of the transition curve into formula (6), and let the constant C' = Jl0. The equation of the cubic parabola transition curve after constant optimization, the curvature radius of any point on the transition curve, and the length from the straight-slow point or the slow-straight point are obtained as follows:

[0065]

[0066] Where: J is the optimization parameter of the cubic parabola transition curve equation, which is calculated using formula (8); C' is a constant, which is the product of parameter J and the required length l0 of the transition curve during the optimization process; x and y are the plane coordinates of any point on the transition curve; ρ and l are the curvature radius of any point on the transition curve and the distance from the straight-slow point or the slow-straight point on the transition curve.

[0067] Example verification:

[0068] Through the above optimization of the cubic parabola transition curve constant, the optimized cubic parabola transition curve related equation is obtained, and the calculation is performed according to the design requirements of the current design specifications. The calculation results are shown in Table 2.

[0069] Table 2 Comparison of actual curvature radius of transition point or transition curve length after optimization with design requirements

[0070]

[0071]

[0072] Through the numerical analysis calculated in Table 2 and considering the engineering accuracy requirements, the linear equation of the optimized cubic parabola transition curve of the present invention can simultaneously meet the requirements that the actual calculated curvature radius of the transition point or the circular transition point is the same as the design requirement radius, and the actual calculated length of the transition curve is consistent with the design requirement length, indicating that the optimized cubic parabola transition curve is smoothly connected to the circular curve, meeting the line smoothness requirements and meeting the safety and comfort requirements of train travel.

Claims

1. A method for optimizing the linear equation of a cubic parabola transition curve for railways and rail transit, comprising the following steps: Step 1: First, meet the required length of the transition curve and calculate the parameter X value ① When the straight-line transition point is taken as the origin and the straight-line side is taken as the x-axis, the linear equation of the cubic parabola transition curve is as follows (1): Where: C is a constant, which is the product of the required circular curve radius R and the required length l0 of the transition curve; x and y are the plane coordinates of any point on the transition curve; h is the superelevation value of any point on the transition curve; and h0 is the superelevation design value of the curve. In order to make the actual calculated length of the cubic parabola transition curve consistent with the design required length, let X be the x-coordinate value of the transition point of the cubic parabola transition curve, and replace l0 in the first line of equation (1) with the parameter X. Then the plane equation of the cubic parabola transition curve evolves as follows: ② Using calculus analysis, let the length of the small segment on the transition curve be dl. According to formula (1), the first-order derivatives of the length of the small segment on the transition curve and the length of the transition curve with respect to x are as follows: ③ Using the original cubic parabola equation, the formula for calculating the radius of curvature of any point on the transition curve is derived through formula (1). The formula for calculating the distance from any point on the transition curve to the straight-slow point or the slow-straight point is derived through formula (4) and by setting C = RX. They are as follows: Where: ρ and l are the curvature radius of any point on the transition curve and the distance from the straight-to-slow point or the slow-to-straight point on the transition curve; ④ Calculate the X value. Let l = l0 and x = X in formula (5). Substitute the required length l0 and required radius R of the transition curve in the design specification into the second line of formula (5) to calculate the X value of each curve. Step 2: Meet the design requirement radius and calculate parameter J ① In order to make the calculated curvature radius of the transition point or the transition point of the cubic parabola transition curve the same as the design requirement radius, replace R in formula (1) with the parameter J, and the cubic parabola transition curve equation evolves as follows: Where: J is the optimization parameter of the cubic parabola transition curve equation, which is close to the design radius; ② From formula (6), the calculation formula for the curvature radius of any point on the transition curve can be evolved into: ③ According to formula (7), with the straight slow point as the origin, the curvature radius of the slow circular point or the circular slow point is calculated as: Where: R' is the calculated value of the curvature radius of the transition point or the transition point of the cubic parabola transition curve; ④ Calculate the J value. Substitute the required length of the transition curve and the X value calculated in step 1 into formula (8), and let R' = R. The parameter J value of each curve can be calculated. Substitute the J value and the required length of the transition curve into formula (6), and let the constant C' = Jl0. The equation of the cubic parabola transition curve after constant optimization, the curvature radius of any point on the transition curve, and the length from the straight-slow point or the slow-straight point are obtained as follows: Where: J is the optimization parameter of the cubic parabola transition curve equation, which is calculated using formula (8); C' is a constant, which is the product of the parameter J and the required length l0 of the transition curve design during the optimization process; x and y are the plane coordinate values ​​of any point on the transition curve, ρ and l are the curvature radius of any point on the transition curve and the length from the straight-to-slow point or the transition-to-straight point on the transition curve.

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