CN110765529A - Railway vertical curve design method based on half-wave sine type and railway ramp - Google Patents
Railway vertical curve design method based on half-wave sine type and railway ramp Download PDFInfo
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Abstract
The invention provides a railway vertical curve design method based on a half-wave sine type, wherein the railway vertical curve is designed according to the following equation: the equation of the vertical curve is as follows:
in the formula (0, L), the intersection point of the linear track and the introduction point of the vertical curve is taken as an origin, the x axis is the extension line direction of the linear track, and the z axis is perpendicular to the horizontal plane; l is the horizontal projection length of the vertical curve, i1、i2The slope of the lead-in end and the slope of the lead-out end of the vertical curve are respectively; wherein z is a coordinate value of a vertical curve connecting the linear track, and the value range of the corresponding rectangular coordinate x is 0-L. The vertical curve line type of the invention can ensure that the curvature of the train is continuous when the train runs at any position of the railway vertical curve, and the wheel rail of the train can not be subjected to sudden impact load, thereby reducing the longitudinal impact on the wheel rail and reducing the abrasion between the rail and the train.
Description
Technical Field
The invention is suitable for traffic road engineering, and can be used for railway lines and highways, in particular to the design of line selection and vertical curves of highways.
Background
With the continuous improvement of the pursuit of people for good life, people have higher requirements on the safety and the comfort of taking a railway train when going out. There are many factors that affect the safety of the vehicle and the comfort of passengers, and the design of the longitudinal section of the railway is one of the important factors. On the longitudinal section of the railway line, the intersection points between the ramp and between the level and the ramp are called grade changing points. When a train runs on a slope change point of two adjacent slopes, the train can suddenly receive vertical positive pressure due to the change of the slope, vertical additional acceleration is generated, strong impact can be generated on wheel tracks of the train, the train vibrates, passengers feel uncomfortable, and hook breakage accidents can be caused if the vertical impact force is too large. In order to ensure the safety of the train and the comfort of passengers, the vertical curve is used for smooth transition at the grade changing point of the railway longitudinal section. The vertical curve is a curve connecting two adjacent slope sections on the line vertical section. The main function of the device is to ease the sudden change of two adjacent ramps on the longitudinal section, and properly combine the device with a flat road or a ramp to reduce the impact generated on the wheel rail. The vertical curve line type commonly used in China at present mainly comprises a parabola and a circular curve, and the difference between the parabola and the circular curve is not great. In the design of railway vertical curves, circular curves are generally adopted because circular curves are more convenient to measure, set and maintain. From a kinetic point of view, if the curve is only continued in shape at the junction (G)0Continuous) and the first derivative is discontinuous, the vehicle will theoretically experience an infinite impact force when passing through that point; if the first derivative of the curve is continuous at the junction point (G)1Continuous) but discontinuous (G) in second derivative2Discontinuous), the vehicle may be subjected to a sudden impact force, causing the vehicle to vibrate. Therefore, the curve has a fundamental defect that the wheel rail is subjected to a sudden impact force to cause vehicle vibration because the acceleration in the vertical direction, which is applied to the train when the train runs, is discontinuous because the curvature of the circular curve is discontinuous although the position and the tangential direction of the connecting point are continuous.
In order to solve the defects of the conventional vertical curve, some researchers have conducted research, wherein a segmental cubic parabola line type is proposed for the Huguan fire, the segmental cubic parabola line type is used for replacing a conventional circular curve, acceleration of an automobile in the vertical direction is gradually changed when the automobile runs on the vertical curve, the mechanical performance of the automobile is improved, but the segmental cubic parabola line type vertical curve does not fundamentally solve the problem of discontinuous acceleration in the vertical direction, impact load can still be generated on a wheel system, vehicle vibration is caused, and the research method aims at a road line. Some scholars also propose the use of cubic interpolation spline curve line type, which can adjust the line shape according to the number of interpolation points to flexibly adapt to the change requirement of control points, but the vertical curve of cubic interpolation spline curve type is also a research for highway lines. At present, the research on the railway vertical curve line type is very little, but the abrasion of the wheel rail caused by the accumulated impact load is very obvious, so the problem needs to be solved urgently. At present, most of domestic tracks are ballastless tracks, the smoothness of the tracks is greatly improved, and the irregularity of a vertical curve becomes one of the main factors influencing the safety and the comfort of a train. Therefore, the invention improves the linear design of the vertical curve, so that the curvature of the connecting point of the vertical curve and the ramp is continuous, and the vertical impact force is fundamentally avoided.
Disclosure of Invention
The invention provides a railway vertical curve design method based on a half-wave sine type, which is used for eliminating vertical impact force generated between a track and a vehicle at a connecting point of a vertical curve, is simple in design method, smooth in curve and capable of being applied to linear design of high-speed railways and high-grade highways.
The invention adopts the following technical scheme:
the invention provides a railway vertical curve design method based on a half-wave sine type, wherein the railway vertical curve is designed according to the following equation:
the equation of the vertical curve is as follows:
in the formula, the intersection point of the linear track and the introduction point of the vertical curve is used as the origin pointThe x axis is the extension line direction of the linear track, and the z axis is vertical to the horizontal plane; l is the horizontal projection length of the vertical curve, i1、i2The slope of the lead-in end and the slope of the lead-out end of the vertical curve are respectively;
wherein z is a coordinate value of a vertical curve connecting the linear track, and the value range of the corresponding rectangular coordinate x is 0-L.
Replacing a vertical curve equation (1) expressed by a rectangular coordinate x with an arc length coordinate l;
(1) wherein x is replaced by the arc length coordinate l:
in the formula, the arc length coordinate l takes the starting point of a vertical curve as an original point and takes the direction along the vertical curve as a positive direction; l is the horizontal projection length of the vertical curve, z is the coordinate value of the vertical curve connecting the linear track, and the value range of the corresponding arc length coordinate L is 0-L.
The method for acquiring the vertical curve comprises the following steps:
setting the second derivative of the vertical curve as follows:
by integrating the above, the first derivative is obtained:
intaglious electricity under the boundary condition of zx＝0＝i1，z′|x＝L＝i2When (2) is substituted into the above formula, the following formula is solved:
the first derivative of the vertical curve is:
integrating the above equation and applying the boundary condition z-x＝0Vertical curve equation is obtained as 0:
the invention also provides a railway ramp, and the vertical curve of the ramp is designed by adopting the railway vertical curve design method.
The curvatures of the vertical curve line type at the lead-in point and the lead-out point are continuous in two orders, so that the vertical acceleration at the starting point and the terminal point of the vertical curve is continuous, the curvatures are continuous when a train runs at any position of the railway vertical curve, the wheel rail of the train cannot be subjected to sudden impact load, the longitudinal impact on the wheel rail is reduced, the abrasion between the rail and the train is reduced, the maintenance of the railway line type is facilitated, the structural life of the wheel rail is prolonged, and the riding comfort of passengers is greatly improved.
Drawings
Fig. 1 shows that in example 1, the running speed of the train is set to be 100m/s, and the acceleration limit is set to be amax＝0.4m/s2The ramp elevation combination graph of (a).
Fig. 2 shows that the running speed of the train is 100m/s and the acceleration limit is a in example 1max＝0.4m/s2Vertical acceleration diagram of (a).
FIG. 3 shows the vehicle speed v is 10m/s and the acceleration limit a is still set as in example 1max＝0.4m/s2A vertical combination graph of (a).
FIG. 4 shows the vehicle speed v is 10m/s and the acceleration limit is a in example 1max＝0.4m/s2Vertical acceleration diagram of (a).
Fig. 5 shows that the running speed of the train is 100m/s and the acceleration limit is a in example 2max＝0.4m/s2A vertical combination graph of (a).
Fig. 6 shows that the running speed of the train is set as v 100m/s and the acceleration limit is set as a in example 2max＝0.4m/s2Vertical acceleration diagram of (a).
FIG. 7 shows the vehicle speed v is 10m/s and the acceleration limit is a in example 2max＝0.4m/s2A vertical combination graph of (a).
FIG. 8 shows the vehicle speed v is 10m/s and the acceleration limit is a in example 2max＝0.4m/s2Vertical acceleration diagram of (a).
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
The invention provides a railway vertical curve design method based on a half-wave sine type, which comprises the following steps:
taking the introduction point of the vertical curve as the origin of coordinates, the horizontal axis as the x axis, the vertical direction as the z axis, and the second derivative of the vertical curve as
In the formula, L is the horizontal projection length of the vertical curve, and A is an unknown constant.
Integrating equation (1) yields:
let the slope of the lead-in ramp of the vertical curve be i1The slope of the leading-out end ramp is i2Then the boundary condition is z'x＝0＝i1，
z′|x＝L＝i2A and C1Is an unknown constant, substituted into the formula (2) and solved
The first derivative of the vertical curve is
Integrate and use formula (3)Using boundary condition z-x＝0Vertical curve equation is obtained as 0:
in a curve coordinate system, replacing a vertical curve equation (1) formula represented by a rectangular coordinate x with an arc length coordinate l, or performing algebraic transformation on the equation (1) to obtain an equation with the same numerical result in engineering application;
(4) wherein x is replaced by the arc length coordinate l:
in the formula, the arc length coordinate l takes the starting point of a vertical curve as an original point and takes the direction along the vertical curve as a positive direction; l is the length of the vertical curve, z is the coordinate value of the vertical curve connecting the linear track, and the value range of the corresponding arc length coordinate L is 0-L.
When the continuity judgment is carried out on the vertical curve, firstly, the second derivative of the vertical curve can be obtained by carrying out derivation on the formula (4):
the continuity of the vertical curve of the present invention is discussed on the basis of the vertical curve and the second derivative as described above: 1. geometric continuity
Z'. non-volatile luminous flux as shown in formula (5)x＝0＝0，z″|x＝LIt can be seen that curvature (second derivative) continuity is achieved at both the entry and exit points of the vertical curve when connected to the linear ramp.
Z'. As shown in formula (3)x＝0＝i1，z′|x＝L＝i2It can be seen that the tangent is continuous at both the entry and exit points of the curve.
Z & lt & gtY & lt & gtx＝0When the value is 0, the introduction point bit pattern continues.
And z is the projection height of the vertical curve in the vertical direction, namely the height difference between the end point of the front ramp and the initial point of the rear ramp.
2. Acceleration limit and curve length
According to the formula (5), when
Time z "has a maximum value:
vertical acceleration a as the vehicle moves through the ramp at a constant velocity vs＝v2z', then the vertical maximum acceleration
From this, a limit value of the length of the vertical curve is obtained, which is determined by the maximum vertical acceleration limit value
Therefore, the curvatures of the vertical curve line type at the leading-in point and the leading-out point are continuous in the second order, so that the vertical acceleration at the starting point and the terminal point of the vertical curve is continuous, and the continuity is known from the analysis of dynamics, the curvatures are continuous when a train runs at any position of the vertical curve of a railway, so that the wheel rail of the vehicle cannot be subjected to sudden impact load, the longitudinal impact on the wheel rail is reduced, the abrasion between the rail and the train is reduced, the maintenance of the railway line type is facilitated, the structural life of the wheel rail is prolonged, and the riding comfort of passengers is greatly improved.
The construction method for constructing the railway ramp by using the vertical curve comprises the following steps:
(1) acquiring parameters of a railway ramp to be constructed, wherein the parameters at least comprise an upper gradient, a lower gradient, a train running speed, a maximum acceleration and the like, and the parameters are given by a railway design party;
(2) after the parameters of the railway ramp are obtained, calculating the corresponding numerical coordinates of each point on the vertical curve of the vertical surface by using the vertical curve equation of the vertical surface given by the formula (4) and taking x as the horizontal coordinate and z as the vertical coordinate according to the slope points of the ramp connected during the actual application of the ramp, wherein the numerical coordinates of a plurality of points form a numerical list;
(3) and (3) according to the design specification requirement and the numerical value list of the specific data acquired in the step (2), applying a coordinate method to design a roadbed, a ballast bed and a laying track, and then laying the track.
In the step (2), the calculation method of the vertical curve and vertical surface combination curve and the numerical list of the vertical acceleration includes, but is not limited to, calculation by an Excel table. When the coordinates of the control points are calculated by using EXCEL, effective numbers above 8 bits can be kept for ensuring the precision so as to reduce the calculation error, and the final coordinate values can reserve the bits according to the precision requirement of the circuit and the actual situation.
When the vertical curve is applied to a high-speed railway or a high-grade highway, due to the fact that the length of the curve is relatively large, the relative difference value of control points is required to be noticed no matter an intelligent total station or a theodolite is used during surveying, setting and work maintenance, and errors caused by long distance measurement are prevented from exceeding a line precision value.
The design, measurement method and construction equipment are the same as the existing traditional vertical curve, and the related calculation can be completed on a personal computer.
Example of engineering design
Example 1: flat-up-flat combination curve
Designing parameters:
the gradient of the uphill slope is set to 0.02.
Vertical curve i of the entry ramp1＝0，i20.02; vertical curve off ramp: i.e. i1＝0.02，i2＝0。
Assuming good train running conditions, the speed is 100m/s, and the acceleration limit is amax＝0.4m/s2According to the formula (7), the horizontal length of the vertical curve should be 785 m.
The length of each of the two horizontal straight roads is 500m, and the length of each of the middle straight roads is 1000 m.
Designing content:
taking the intersection point of the introduction points of the linear track and the vertical curve as an origin, the extension line of the linear track is an x axis, the vertical direction is a z axis, a control point is arranged every 10m, and i1、i2And the values of L are respectively substituted into the formulas (4) and (5), so that a vertical surface combination curve and a vertical acceleration can be respectively obtained, as shown in fig. 1 and fig. 2. Since the number points are many, not all are listed. On a straight ramp section, the vertical coordinate z value and the vertical acceleration asThe variation is very small or zero, so start and end points are enumerated; in the vertical curve segment, a coordinate point is listed every 40m, the number of control points can be arranged according to actual requirements when engineering is implemented, and the following engineering example is similar to the example. The specific numerical values are listed in tables 1 and 2.
| table 1 numerical value list unit m of vertical face combination curve
TABLE 2 vertical acceleration values tabulated units m/s2
If the vehicle speed is set to be 10m/s and the acceleration limit is still a near the station yardmax＝0.4m/s2According to the formula (7), the horizontal length of the vertical curve is 7.85m, and L is 10 m. The length of each of the two horizontal straight roads is 5m, the length of the middle straight road is 10m, the vertical combined curve and the vertical acceleration are shown in figures 3 and 4, and the method for calculating the numerical points is the same as that of the combined curve.
Example 2: up-down (down-up) ramp combination curve
Designing parameters:
up-down ramp: slope of uphill slope is taken i10.02, the slope of the downhill slope is taken as i2＝0.02；
Down-up ramp: i.e. i1-0.02, slope of downhill i2＝0.02；
Assuming good driving conditions, the vehicle speed is 100m/s, and the acceleration limit is still set as amax＝0.4m/s2The length of the vertical curve is taken as 1500m, and the slopes at two ends are respectively taken as 1000 m.
Designing content:
taking the intersection point of the introduction points of the linear track and the vertical curve as an origin, the extension line of the linear track is an x axis, the vertical direction is a z axis, a control point is arranged every 10m, and i1、i2And the values of L are respectively substituted into the formulas (4) and (5), so that a vertical surface combination curve and a vertical acceleration can be respectively obtained, as shown in fig. 5 and 6. The specific numerical values are listed in tables 3 and 4, and the value of the z-axis and the vertical acceleration of the up-down ramp combined curve are respectively recorded as z1(m)、a1(m/s2) The combined curves of the down-up ramps are respectively denoted as z2(m)，a2(m/s2)。
Table 3 vertical combination curve value list: unit mm
Table 4 vertical acceleration values list: unit m/s2
Example 3: up-up ramp combination curve
Designing parameters:
slope of uphill slope is taken i1Connecting ramp i 0.02 ═ d2＝0.01；
Ascending ramp i1Up slope i of 0.012＝0.02；
Assuming good running conditions of the train, the speed is 100m/s, and the acceleration limit is amax＝0.3m/s2The length of the vertical curve is 500m, and the slopes at two ends are 500m respectively.
Designing content:
the intersection point of the linear track and the introduction point of the vertical curve is taken as the original pointPoint, the extension line of the linear track is an x axis, the vertical direction is a z axis, a control point is arranged every 5m, and i is1、i2And the values of L are respectively substituted into the formulas (4) and (5), so that a vertical surface combination curve and a vertical acceleration can be respectively obtained, as shown in fig. 7 and 8. The numerical values are tabulated in tables 5 and 6, the upward gradient i1Connecting ramp i 0.02 ═ d2The value of the z-axis and the vertical acceleration of the combined curve of 0.01 are respectively denoted as z3(m)、a3(m/s2) Ascending ramp i1Up slope i of 0.012The value of the z-axis and the vertical acceleration of the combined curve, respectively, are denoted as z, 0.024(m)、a4(m/s2)。
Table 5 vertical combination curve value list: unit m
Table 6 vertical acceleration values list: unit m/s2
In view of the accuracy requirements, the use of an ES100 intelligent total station or an LP400 laser electronic theodolite is recommended at the time of survey.
In the description herein, particular features, structures, materials, or characteristics may be combined in any suitable manner in any one or more embodiments or examples.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the present invention should be covered by the present invention.
Claims (4)
1. A railway vertical curve design method based on half-wave sine type is characterized in that: the railway vertical curve is designed according to the following equation:
the equation of the vertical curve is as follows:
in the formula, the intersection point of a straight line track and a vertical curve introduction point is taken as an original point, the x axis is the extension line direction of the straight line track, and the z axis is perpendicular to the horizontal plane; l is the horizontal projection length of the vertical curve, i1、i2The slope of the lead-in end and the slope of the lead-out end of the vertical curve are respectively;
wherein z is a coordinate value of a vertical curve connecting the linear track, and the value range of the corresponding rectangular coordinate x is 0-L.
2. The method for designing the railway vertical curve based on the half-wave sine type according to the claim 1, which is characterized in that:
a vertical curve equation (1) expressed by an arc length coordinate/a substituted rectangular coordinate x;
(1) in which the arc length coordinates/substitution x:
in the formula, the arc length coordinate/the starting point of the vertical curve is taken as the original point, and the direction along the vertical curve is the positive direction; l is the horizontal projection length of the vertical curve, z is the coordinate value of the vertical curve connecting the linear track, and the value range of the corresponding arc length coordinate/is 0-L.
3. The method for designing the railway vertical curve based on the half-wave sine type according to the claim 1, which is characterized in that:
the method for acquiring the vertical curve comprises the following steps:
setting the second derivative of the vertical curve as follows:
by integrating the above, the first derivative is obtained:
intaglious electricity under the boundary condition of zx＝0＝i1，z′|x＝L＝i2When (2) is substituted into the above formula, the following formula is solved:
the first derivative of the vertical curve is:
integrating the above equation and applying the boundary condition z-x＝0Vertical curve equation is obtained as 0:
4. a railway ramp characterized by: the vertical curve of the ramp is designed by the method for designing the vertical curve of the railway according to any one of claims 1-2.
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