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The paper presents an analytical method of identifying the curvature of the turnout diverging track consisting of sections of varying curvature. Both linear and nonlinear (polynomial) curvatures of the turnout diverging track are identified and evaluated in the paper. The presented method is a universal one; it enables to assume curvature values at the beginning and end point of the geometrical layout of the turnout. The results of dynamics analysis show that widely used in railway practice, clothoid sections with nonzero curvatures at the beginning and end points of the turnout lead to increased dynamic interactions in the track-vehicle system. The turnout with nonlinear curvature reaching zero values at the extreme points of the geometrical layout is indicated in the paper as the most favourable, taking into account dynamic interactions occurring in the track-vehicle system.

Typical, used since the beginning of railway engineering, geometrical layout of the turnout diverging track consists of a single circular arc without transition curves. It introduces sudden, abrupt changes of the horizontal curvature of the layout at the beginning and end of the turnout diverging track, which increases dynamic interactions in the track-vehicle system, particularly unfavourable in high speed rail (HSR). Investigation and evaluation of geometrical layouts of the turnout diverging track are still a current issue.

Recently, aiming at smoothing changes of the curvature at the neuralgic regions of the turnout diverging track, the clothoid sections have been introduced at both sides of the circular arc [

In the turnout with linear curvature sections, a diverging track is divided into three zones (

・ a beginning zone of the length

・ a middle zone of the length

・ an end zone of the length

The various values of curvature and length of each section can be applied in the designing process. Curvature of the turnout diverging track is described by an analytical function

This paper presents the identification of analytical functions

In this paper, the Cartesian coordinates of the turnout diverging tracks are not presented. The method of the identification of the Cartesian coordinates from the curvature

In the beginning zone of the turnout the considered issue is identified by boundary

conditions [

and a differential equation

After determining the constants, the solution of the differential problem (1), (2) is as follows:

The slope of the tangent at the end of the zone, for

In the circular arc zone, i.e. for

At the end of circular arc the slope of the tangent is defined by the formula:

In the end zone of the turnout the following boundary conditions are adopted:

for the differential Equation (2). After determining the constants, the solution of the differential problem (2), (7) is as follows:

The slope of the tangent at the end of the turnout is defined by the formula:

from which the turnout angle

The curvature of the turnout diverging track in

The following boundary conditions have been adopted:

to the differential equation

with assumption, that coefficient

As a result of solving the differential problem (11), (12) the following curvature has been obtained:

Function

At the end of the zone, for

Similarly to the middle zone described in the section 2.2, i.e. for

Assuming the boundary conditions:

for the differential Equation (12) the following solution has been obtained:

where

Assuming C = 1.5 the following coefficient formulas have been obtained:

The slope of the tangent at the end of the turnout, for

In

In order to ensure a reliable comparative analysis of the geometrical layouts presented in

・ the turnout angle 1:n , where n = 50,

・ the curvature values

・ the circular arc radius

・ the length of the beginning zone

・ the length of the circular arc

No | Curvature | |||||||
---|---|---|---|---|---|---|---|---|

1 | Constant | 1/6000 | 0.00 | 1/6000 | 119.984 | 0.00 | 1/6000 | 119.984 |

2 | Linear | 1/16000 | 40.00 | 1/6000 | 64.584 | 45.00 | 1/25000 | 149.584 |

3 | Linear | 1/16000 | 40.00 | 1/6000 | 62.484 | 60.00 | 0 | 162.484 |

4 | Linear | 0 | 60.00 | 1/6000 | 59.984 | 60.00 | 0 | 179.984 |

5 | Nonlinear | 1/16000 | 40.00 | 1/6000 | 57.184 | 45.00 | 1/25000 | 142.184 |

6 | Nonlinear | 1/16000 | 40.00 | 1/6000 | 51.859 | 60.00 | 0 | 151.859 |

7 | Nonlinear | 0 | 60.00 | 1/6000 | 44.984 | 60.00 | 0 | 164.984 |

The highest velocity on a circular arc without superelevation (i.e. in the middle zone) results from the following condition:

while in the extreme zones the condition is as follows

where:

V―train velocity [km/h],

R―circular arc radius [m],

^{2}],

^{2}],

^{3}].

^{3}].

It is assumed that

On a circular arc without transition curves (turnout 1) the acceleration changes linearly from 0 to

with condition

the limit of the velocity

On a circular arc in the middle zone with sections of changing curvature in the beginning and end zones the limit of the velocity is described by the formula

In the beginning zone where curvature changes linearly a rate of acceleration changes

from which the minimal length

Nonlinear curvature (polynomial) induces changing rate of acceleration changes

An increase by 50% of the limit value

For the assumed turnout angle 1:50 (i.e. n = 50) the following slope of the tangent has been obtained, using Equation (10):

Assuming

The geometrical parameters of the selected seven turnouts are presented in

and in the Equation (19) for nonlinear curvature:

The function of lateral acceleration

With increased speed requirements on railways, the dynamic effects minimization is a current issue, especially in HSR. Basing on the assumption that horizontal curvature changes are a forcing factor of the lateral oscillations, selected seven geometrical layouts of the turnout diverging track are compared in terms on their impact on the dynamic interactions occurring in a rail-vehicle system. In the presented comparative analysis of the layouts, structural aspects of the rail

No | Beginning zone | Middle zone | End zone |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 |

vehicle are omitted.

A dynamic model with one degree of freedom, consisting of a mass with a spring and a damper is applied to compare the dynamic interactions occurring on the various turnout diverging tracks. An additional parameter―a length of the rigid base of a wagon has been introduced, which results in referring to the lateral acceleration of the wagon mass center (arithmetic mean of accelerations occurring in the front and rear bogies).

The lateral acceleration

where:

D―Lehr’s damping coefficient,

Lehr’s damping coefficient D is used as a damping measure in the railway engineering. In the presented paper D = 0.175 and ω = 3.5 π/s are assumed. The assumed value of D has been obtained in the experimental research presented in [

The function of oscillations

where:

are assumed as criteria of the dynamic effects evaluation presented in Section 6.

The length of the rigid base (it has been assumed

1).

The acceleration of oscillating motion

Apart from the beginning and end zones of the turnout diverging track, the dynamic interactions occur also at the beginning and end of the middle zone, as shown in

The comparative analysis of the selected layouts of the turnouts diverging track has been carried out using dynamic indicators:

As shown evidently in

No | Zones of dynamic effects along the turnout diverging track | |||||||
---|---|---|---|---|---|---|---|---|

Beginning zone | Beginning of the circular arc | End of the circular arc | End zone | |||||

^{2}/s^{2}] | ^{2}] | ^{2}/s^{2}] | ^{2}] | ^{2}/s^{2}] | ^{2}] | ^{2}/s^{2}] | ^{2}] | |

1 | 216.89 | 174.40 | 0.00 | 0.000 | 0.00 | 0.00 | 216.90 | 174.40 |

2 | 329.71 | 266.95 | 3.83 | 2.932 | 4.19 | 3.23 | 209.19 | 166.86 |

3 | 329.71 | 266.95 | 3.83 | 2.932 | 3.81 | 2.80 | 3.81 | 2.80 |

4 | 4.08 | 3.13 | 5.61 | 4.710 | 4.14 | 3.19 | 4.14 | 3.19 |

5 | 330.84 | 268.06 | 0.10 | 0.055 | 0.11 | 0.06 | 209.24 | 165.64 |

6 | 330.84 | 268.06 | 0.11 | 0.060 | 0.05 | 0.01 | 5.91 | 4.35 |

7 | 6.10 | 4.69 | 0.60 | 0.030 | 1.87 | 0.09 | 6.40 | 4.72 |

The assumption of

along the whole turnout diverging track; it is concerned layouts with sections of linear curvature (turnout 4―

The presented results leads to conclusion that widely applied in a railway practice “clothoid sections” with curvatures

slightly increase (

The acceleration in oscillating motion

Taking into account the dynamic properties and the length of the layout, the turnout diverging track 7 is definitely the most favourable. Turnout 7 in comparison with turnout 4 has better dynamic properties in the middle zone, shorter length and insignificantly worse values of dynamic indicators in the beginning and end zones (

As a result of dynamics analysis it has been proved that the most favourable dynamic properties can be achieved by applying a nonlinear curvature in the beginning and end zones of the turnout diverging track and assuming zero curvature value at the extreme points of the geometrical layout.

Assuming

・ in the beginning zone, for

・ in the middle zone, i.e. for

・ in the end zone, for

The slope of the tangent at the end of the turnout, for

In

Typical turnout diverging track consists of a single circular arc without transition curves. It introduces sudden, abrupt changes of the horizontal curvature of the layout at the beginning and end of the turnout diverging track, which increases dynamic interactions in the track-vehicle system, particularly unfavourable in HSR.

The paper presents a universal, analytical method of identifying the curvature of the turnout diverging track. Both linear and nonlinear (polynomial) curvatures of the turnout diverging track are identified and evaluated using a dynamic

model. The presented method enables to assume the curvature values at the beginning and end point of the geometrical layout of the turnout. The length of the circular arc is adjusted to obtain the assumed turnout angle.

Recently, aiming at smoothing changes of the curvature at the neuralgic regions of the turnout diverging track, the clothoid sections have been introduced at both sides of the circular arc. The curvature of the applied clothoid sections changes linearly but in many cases does not reach zero value at the extreme points (i.e. at the beginning and end points of the turnout). The results of dynamics analysis presented in the paper show that clothoid sections with nonzero curvature at the beginning and end points of the turnout lead to increased dynamic interactions in the track-vehicle system. Dynamic interactions can be decreased by applying curvature reaching zero at the extreme points of the turnout.

The paper presents the evaluation of the selected seven geometrical layouts of the turnout diverging track and indicates the most favourable solution for HSR. The most favourable from the dynamic properties point of view is the turnout diverging track with nonlinear curvature reaching zero values at the extreme points of the turnout.

Koc, W. and Palikowska, K. (2017) Dynamic Analysis of the Turnout Diverging Track for HSR with Variable Curvature Sections. World Journal of Engineering and Technology, 5, 42-57. https://doi.org/10.4236/wjet.2017.51005