### Method of construction spatial transition curve

S. A. Ustenko, S. V. Didanov

#### Abstract

Purpose. The movement of rail transport (speed rolling stock, traffic safety, etc.) is largely dependent on the quality of the track. In this case, a special role is the transition curve, which ensures smooth insertion of the transition from linear to circular section of road. The article deals with modeling of spatial transition curve based on the parabolic distribution of the curvature and torsion. This is a continuation of research conducted by the authors regarding the spatial modeling of curved contours. Methodology. Construction of the spatial transition curve is numerical methods for solving nonlinear integral equations, where the initial data are taken coordinate the starting and ending points of the curve of the future, and the inclination of the tangent and the deviation of the curve from the tangent plane at these points. System solutions for the numerical method are the partial derivatives of the equations of the unknown parameters of the law of change of torsion and length of the transition curve. Findings. The parametric equations of the spatial transition curve are calculated by finding the unknown coefficients of the parabolic distribution of the curvature and torsion, as well as the spatial length of the transition curve. Originality. A method for constructing the spatial transition curve is devised, and based on this software geometric modeling spatial transition curves of railway track with specified deviations of the curve from the tangent plane. Practical value. The resulting curve can be applied in any sector of the economy, where it is necessary to ensure a smooth transition from linear to circular section of the curved space bypass. An example is the transition curve in the construction of the railway line, road, pipe, profile, flat section of the working blades of the turbine and compressor, the ship, plane, car, etc.

#### Keywords

modeling; spatial transition curve; parabolic distribution of curvature and torsion; railways; road safety

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#### References

Amelin S.V., Danovskiy L.M. Put i putevoye khozyaystvo [Road and Track Facilities]. Мoscow, Тransport Publ., 1986. 215 с.

Yelfimov G.V. Teoriya perekhodnykh krivykh [Theory of transitional curves]. Moscow, Transzheldorizdat Publ., 1948. 311 с.

Lahuta V.V. Udoskonalennia proektuvannia kryvykh zaliznychnoi kolii v plani. Avtoreferat Diss. [Im-provement of railway curves designing in plan. Author’s abstract.]. Dnipropetrovsk, 2002, 18 p.

Finitskiy S.I. Put i putevoye khozyaystvo zheleznykh dorog SShA [Road and Track Facilities of USA Railways]. Moscow, Transport Publ., 1987. 215 с.

Rusu S.P., Kravets V.V. Matematicheskaya model puti prostranstvennoy konfiguratsii pri razlichnykh rezhimakh dvizheniya transportnykh ekipazhey [Mathematical model of the spatial configuration of the way at various modes of traffic crews]. Zbirnyk naukovykh prats Dnipropetrovskoho derzhavnoho tekhnichnoho universytetu zaliznychnoho transportu “Transport. Matematychne modeliuvannia v inzhenernykh ta ekonomichnykh zadachakh transportu” [Proc. of the State Technical University of Railway Transport Transportation. Mathematical modeling in engineering and business problems of transport], 1999, pp.114-119.

Ustenko S.A., Didanov S.V. Heometrychne modeliuvannia perekhidnoi kryvoi u prostori [Geometric modeling of the ease curve in scope]. Prykladna heometriia ta inzhenerna hrafika - Applied Geometry and Engineering Graphics, 2012, issue 89, pp. 368-372.

Ustenko S.A. Heometrychne modeliuvannia prostorovykh kryvykh linii zadanykh kryvyny ta skrutu [Geometric modeling of spatial curves given curvature and torsion] Heometrychne ta kompiuterne modeliuvannia. – Geometric and computer modeling, 2011, issue 29, pp. 86-90.

Zhaia W.M., Wanga K.Y. Lateral interactions of trains and tracks on small-radius curves. Vehicle System Dynamics, 2006, vol. 44, pp. 520-530.

#### GOST Style Citations

1. Амелин, С. В. Путь и путевое хозяйство / С. В. Амелин, Л. М. Дановский. – М. : Транспорт, 1986. – 215 с.

2. Ельфимов, Г. В. Теория переходных кривых / Г. В. Ельфимов. – М. : Трансжелдориздат, 1948. – 311 с.

3. Лагута, В. В. Удосконалення проектування кривих залізничної колії в плані : автореф. дис. …канд. техн. наук : 05.22.06 / Лагута Василь Васильович ; "Залізнична колія". – Д., 2002. – 18 с.

4. Финицкий, С. И. Путь и путевое хозяйство железных дорог США / С. И. Финицкий. – М. : Транспорт, 1987. – 215 с.

5. Русу, С. П. Математическая модель пути пространственной конфигурации при различных режимах движения транспортных экипажей / С. П. Русу, В. В. Кравец // Транспорт. Математичне моделювання в інженерних та економічних задачах транспорту : зб. наук. праць / Дніпропетр. державний техн. ун-т залізн. трансп. – Д., 1999. − С. 114−119.

6. Устенко, С. А. Геометричне моделювання перехідної кривої у просторі / С. А. Устенко, С. В. Діданов // Прикладна геометрія та інженерна графіка. – К. : КНУБА, 2012. – Вип. 89. – С. 368−372.

7. Устенко, С. А. Геометричне моделювання просторових кривих ліній заданих кривини та скруту / С. А. Устенко // Геометричне та комп’ютерне моделювання. – Х. : ХДУХТ, 2011. – Вип. 29. – С. 86−90.

8. Zhaia, W. M. Lateral interactions of trains and tracks on small-radius curves / W. M. Zhaia, K. Y. Wanga // Vehicle System Dynamics. − Vol. 44, Supplement. − 2006. − Р. 520−530.

DOI: https://doi.org/10.15802/stp2013/11394

### Cited-by:

1. MODELING THE TRANSITION CURVE ON A LIMITED TERAIN
V. D. Borisenko, S. A. Ustenko
Science and Transport Progress. Bulletin of Dnipropetrovsk National University of Railway Transport  Issue: 2(68)  First page: 92  Year: 2017
doi: 10.15802/stp2017/99942