Polynomial Solutions of Transition Curves

Kobryń, Andrzej

doi:10.1061/(asce)su.1943-5428.0000044

Cited by 28 publications

(20 citation statements)

References 7 publications

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“…Solving the optimization problem determines the polynomials that simulate transition curves, which provide comfortable conditions for passengers. Polynomial curves are also used in [8] to construct a curvilinear section applying two transition curves and one circular curve. The same curves are used to construct vertical arches of transition curves [9].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Devising an approach to the geometric modeling of railroad tracks along curvilinear sections

Borisenko

Ustenko

2022

EEJET

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This paper proposes an approach to arranging curvilinear sections of the railroad track, which involves the replacement of two transitional ones and a circular curve with one curvilinear section. Modeling this section implies the construction of a curve in the plan and profile, the joint consideration of which forms the spatial appearance of the curve, which ensures a smooth transition between the rectilinear rails. The need for such transitions is due to the terrain, the need to bypass settlements, and the presence of geological, geographical, and other obstacles that occur when laying railroads. A curvilinear section is modeled in the plan using a curve that is represented in natural parameterization under the law of curvature distribution in the form of a fifth-power polynomial. At the same time, at the start and end points of the section, the curvature and its derivative accept zero values. The outer rail elevation retraction (modeling in the profile) is performed using the curve built, whose sections are also represented in natural parameterization with the dependences of the curvature distribution on the arc length in the form of fourth-power polynomials. At the docking point of the sections, the third order of smoothness is ensured, which implies the equality of values of the functions, their derivatives, the curvature, and a derivative of the curvature from the length of its natural arc. Measures have been proposed to ensure the predefined track gauge retraction. The application of the proposed approach to model a railroad track along the curvilinear section with a variable-radius curve could make it possible to achieve a favorable curvature distribution, a smooth elevation retraction of the outer rail and the track gauge. That would consequently improve the safety of rolling stock running along the curvilinear section of the track, reduce the lateral and vertical efforts that predetermine the wear of rails and wheelsets.

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Devising an approach to the geometric modeling of railroad tracks along curvilinear sections

Borisenko

Ustenko

2022

EEJET

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“…Other designing parameters for these curves are values tanα, tanuP, and tanuK. In the case of the solution (2), obtaining of the curvature distribution, which is appropriate for S-shaped transition curves, requires to meet of following conditions (in agreement to Kobryń (2017a) and allow obtaining a correct distribution of curvature within the curve (Kobryń, 2011).…”

Section: S-shaped Transition Curves As An Element Of Reverse Curves Imentioning

confidence: 99%

S-Shaped Transition Curves as an Element of Reverse Curves in Road Design

Kobryń

Stachera

2019

BJRBE

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A road designing involves horizontal and vertical alignment. The horizontal geometry is formed by straight and curvilinear sections that are traditionally formed using circular and transition curves (mainly the clothoid). Different geometric systems that are designed using circular and transition curves are between others circular curves with symmetrical or unsymmetrical clothoids, combined curves, oval curves and reverse curves. Designing these systems is quite complex. Therefore, so-called S-shaped transition curves are an alternative to traditional approaches. These curves are known from literature and are modern geometric tools for the shaping of reverse curves. The paper analyses the basic geometric properties of these curves as well as compare to the geometry of the appropriate geometric systems, which are formed with clothoid or using S-shaped transition curves. In addition, a procedure for designing reverse curves using S-shaped transition curves was proposed. Another research topic was the comparison of the analysed reverse curves (created using polynomial transition curves) with traditional curves (created using the clothoid). The results of the studies, despite the noticeable differences in the geometry of the compared components, confirm the practical usefulness of the S-shaped transition curves for designing the geometry of the route.

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“…The development of a method for precise determination of route coordinates seems to be the most appropriate course of action (Ahmad, Ali 2008;Bosurgi, D' Andrea 2012;Bosurgi et al 2016;Kobryń 2011Kobryń , 2014Koc 2014;Ziatdinov et al 2012aZiatdinov et al , 2012b. This paper provides the examples of such an approach, which radically diverges from the traditional approach.…”

Section: Introductionmentioning

confidence: 99%

Modelling of joining route segments of different curvature

Koc

Palikowska

2016

The Baltic Journal of Road and Bridge Engineering

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The paper presents a new general method of modelling route segments curvature using differential equations. The method enables joining of route segments of different curvature. Transitional curves of linear and nonlinear curvatures have been identified in the case of joining two circular arcs by S-shaped and C-oval transitions. The obtained S-shaped curves have been compared to the cubic C-Bezier curves and to the Pythagorean hodograph quantic Bezier curve using the Lateral Change of Acceleration diagram and the dynamic model. The analysis of dynamic properties has showed an advantage of the obtained transition curve of nonlinear curvature over Bezier curves.

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Polynomial Solutions of Transition Curves

Cited by 28 publications

References 7 publications

Devising an approach to the geometric modeling of railroad tracks along curvilinear sections

Devising an approach to the geometric modeling of railroad tracks along curvilinear sections

S-Shaped Transition Curves as an Element of Reverse Curves in Road Design

Modelling of joining route segments of different curvature

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