Optimization method for linear equation of cubic parabola easement curve of railway and rail transit

Abstract

The invention discloses an optimization method for a cubic parabola easement curve linear equation of railway and rail transit, so as to ensure that the actual calculation length of an easement curve is consistent with the length required by design, the calculation curvature radius of a slow point or a circle slow point is consistent with the radius required by design, and smooth connection between a circular curve and the easement curve is ensured. The construction and operation safety of rail transit such as railways is ensured, and the riding comfort is improved. The method includes the following steps that firstly, the length required by easement curve design is met, and a parameter X value is calculated; 2, a parameter J is calculated when the design requirement radius is met; cubic parabola easement curve equation after constant optimization, and calculation formula of curvature radius of any point on easement curve and length from straight-slow point or slow-straight point as follows: # imgabs0 #

Description

Technical Field

[0001] The 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 transit. Background Art

[0002] my country requires that railways and wheel-rail rail transit should use cubic parabolas as transition curves, and smoothly connect circular curves and straight lines to form route design lines. However, during design and surveying, according to the current equation of my country's cubic parabola transition curve, it was found that the actual calculated length of the transition curve and the design required length, the curvature radius of the transition point or the calculated curvature radius of the circular transition point and the design required radius cannot be the same at the same time, resulting in the inability to smoothly close the circular curve and the transition curve, which will affect the safety and smoothness of the line operation and cannot be used for design, construction, maintenance and repair. Research on the method of optimizing the constants of the current cubic parabola transition curve equation and proposing the correct cubic parabola transition curve equation is very important for the line design, construction, maintenance and software development of railways and various types of rail transit using cubic parabola transition curves.

[0003] In the design, the calculated length of the transition curve, the transition point or the calculated curvature radius of the circular transition point are the actual design values. When the cubic parabola is used as the transition curve, when it is calculated and designed according to the current equation, if the transition curve length is consistent with the design required length, the actual calculated curvature radius of the transition curve transition point or circular transition point is different from the design required radius; on the contrary, if the calculated curvature radius of the transition curve transition point or circular transition point is forced to be the same as the design required radius, the actual calculated length of the transition curve is inconsistent with the design required length, that is, the actual calculated curvature radius of the transition point or circular transition point and the transition curve length cannot meet the design required parameters at the same time. That is, the difference between the calculated value of the original cubic parabola transition curve equation and the design required value is very large, resulting in the cubic parabola transition curve and the circular curve cannot be smoothly closed according to the design required parameters, which has a great impact on the safety and smoothness of the line. 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 a 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, ensure the smooth connection between the circular curve and the transition curve, ensure the safety of construction and operation of railways and other rail transit, improve riding comfort, and can be used in engineering for railway and rail transit line design, surveying, construction, overhaul, software development and theoretical research.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows:

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

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

[0008] ① When the straight-slow 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] In the formula: C is a constant, which is the product of the circular curve radius R required by the design and the length l0 required by the transition curve design, x and y are the plane coordinate values ​​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), and 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 C=RX, which are as follows:

[0017]

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

[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 circular transition point of the cubic parabola transition curve the same as the design required 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), taking the straight-slow point as the origin of the rectangular coordinate system, the curvature radius of the slow-circle point or the round-slow 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 radius of curvature of any point on the transition curve, and the length calculation formula 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 design in the optimization process; x, y are the plane coordinate values ​​of any point on the transition curve, ρ, l are the curvature radius of any point on the transition curve, and the length of the transition curve from the straight-slow point or the slow-straight point.

[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 the compilation of design software for transportation projects such as railways, rail transit and roads that use cubic parabolas as transition curves. The present 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 is further described below in conjunction with the accompanying drawings and embodiments.

[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 requirement values ​​is shown in the following table.

[0036] Through calculation based on formula (5) and the design requirement value of the current specification, it is found that when the cubic parabola equation originally specified 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 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 circular transition point is the same as the design requirement radius, the actual calculated length of the transition curve is different from the design requirement length. That is, the actual calculated curvature radius of the transition point or 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] According to Table 1, the calculated value of the original cubic parabola transition curve equation is very different from the design requirement value, which results in the cubic parabola transition curve and the circular curve being unable to be smoothly closed according to the design requirement parameters, which has a great impact on the safety and smoothness of the line. In order to meet the requirements of the length and geometric shape of the cubic parabola transition curve in design, measurement and construction, and maintenance and repair, and to improve the safety and smoothness of line operation, the constant C value of the original cubic parabola plane equation must be optimized.

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

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

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

[0043] ① When the straight-slow 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 national "Eighth Five-Year Plan" scientific and technological key project "Research on the Selection of Design Parameters for Bridges and Tunnels of High-Speed ​​Railway Lines" "Preliminary Study on the Parameters of Plane and Elevation Curves of High-Speed ​​Railways", which is the current design specification requirement for railways and various types of rail transit in my country.

[0046] In the formula: C is a constant, which is the product of the circular curve radius R required by the design and the length l0 required by the transition curve design, x and y are the plane coordinate values ​​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), and 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 C=RX, which are as follows:

[0052]

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

[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 circular transition point of the cubic parabola transition curve the same as the design required 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), taking 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 radius of curvature of any point on the transition curve, and the length calculation formula 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 design in the optimization process; x, y are the plane coordinate values ​​of any point on the transition curve, ρ, l are the curvature radius of any point on the transition curve, and the length of the transition curve from the straight-slow point or the slow-straight point.

[0067] Example verification:

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

[0069] Table 2 Comparison of actual curvature radius of the transition point or transition point after optimization and the length of the transition curve with the 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 required radius, and the actual calculated length of the transition curve is consistent with the design required length, indicating that the optimized cubic parabola transition curve is smoothly connected with the circular curve, meeting the line smoothness requirements and meeting the safety and comfort requirements of train travel.

Claims

1. The optimization method of the linear equation of the cubic parabola transition curve of railway and rail transportation includes the following steps: Step 1: First meet the required length of the transition curve design and calculate the parameter X value ① When the straight-slow 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): In the formula: C is a constant, which is the product of the circular curve radius R required by the design and the length l0 required by the transition curve design, x and y are the plane coordinate values ​​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), and 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 C=RX, which are as follows: Where: ρ, l are the radius of curvature of any point on the transition curve, and the length of the transition curve from the straight-to-slow point or the slow-to-straight point; 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 circular transition point of the cubic parabola transition curve the same as the design required 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), taking 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 radius of curvature of any point on the transition curve, and the length calculation formula 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 parameter J and the required length l0 of the transition curve design in the optimization process; x and y are the plane coordinate values ​​of any point on the transition curve, ρ and l are the radius of curvature of any point on the transition curve and the length from the straight-to-slow point or the slow-to-straight point on the transition curve.

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