Modelling of the shape of railway transition curves from the point of view of passenger comfort




high-speed rail lines, railway transition curves, passenger comfort, computer simulation, optimisation


In the past, railway transition curves were not used. Instead of it, a simple connection of the straight track and circular arc was applied. Nowadays, such simplicity is not allowed due to the increasing vehicle operating velocities. It is mainly visible in the high-speed train lines, where long curves are used. The article aims to develop a new shape of railway transition curves for which passenger travel comfort will be as high as possible. Considerations in this paper concern the polynomials of 9th- and 11th-degrees, which were adopted to the mathematical model of the mentioned shape of curves. The study's authors applied a 2-axle rail vehicle model combined with mathematically understood optimisation methods. The advanced vehicle model can better assign the dynamical properties of railway transition curves to freight and passenger vehicles. The mentioned model was adopted to simulate rail vehicle movement in both cases of the shape of transition curves and the shape of circular arc (for comparison of the results). Passenger comfort, described by European Standard EN 12299, was used as the assessment criterion. The work showed that the method using the 2-axle railway vehicle model combined with mathematically understood optimisation works correctly, and the optimisation of the transition curve shape is possible. The current study showed that the 3rd-degree parabola (the shape of the curve traditionally used in railway engineering) is not always the optimum shape. In many cases (especially for the long curves), the optimum shape of curves is between the standard transition curves and the linear curvature of the 3rd-degree parabola. The new shapes of the railway transition curves obtained when the passenger comfort is taken into account result in new railway transition curves shapes. In the authors' opinion, the results presented in the current work are a novelty in optimisation and the properties assessment of railway transition curves.


Ahmad, A., & Ali, J. M. (2008). G3 transition curve between two straight lines. In: 2008 Fifth International Conference on Computer Graphics, Imaging and Visualisation (pp. 154-159). IEEE.

Barna, Z., & Kisgyörgy, L. (2015). Analysis of Hyperbolic Transition Curve Geometry. Periodica Polytechnica Civil Engineering, 59(2), 173–178. doi: 10.3311/PPci.7834.

CEN (2009). Railway applications – ride comfort for passengers – measurement and evaluation (EN 12299). Brussels: CEN.

Commission Regulation (EU) No 1299/2014 of 18 November 2014 on the technical specifications for interoperability relating to the ‘infrastructure’ subsystem of the rail system in the European Union (Text with EEA relevance).

Eliou, N., & Kaliabetsos, G. (2014). A new, simple and accurate transition curve type, for use in road and railway alignment design. Eur. Transp. Res. Rev. 6, 171–179. doi: 10.1007/s12544-013-0119-8.

Esveld, C. (2001). Modern railway track (Vol. 385). Zaltbommel: MRT-productions.

Fischer, S. (2009). Comparison of railway track transition curves. Pollack Periodica, 4(3), 99-110.

Gołębiowski, P., & Kukulski, J. (2020). Preliminary study of shaping the railway track geometry in terms of their maintenance costs and capacity. Archives of Transport, 53(1), 115-128. DOI:

Gołębiowski, P., Żak, J., & Jacyna-Gołda, I. (2020). Approach to the Proecological Distribution of the Traffic Flow on the Transport Net-work from the Point of View of Carbon Dioxide. Sustainability, 12(17), 6936. doi:10.3390/su12176936

Izdebski, M., & Jacyna, M. (2021). An Efficient Hybrid Algorithm for Energy Expenditure Estimation for Electric Vehicles in Urban Service Enterprises. Energies, 14(7), 2004. doi:10.3390/en14072004.

Jacyna, M., & Krześniak, M. (2017). Computer support of decision-making for the planning movement of freight wagons on the rail network. In Scientific And Technical Conference Transport Systems Theory And Practice (pp. 225-236). Springer, Cham.

Jacyna-Gołda, I., Żak, J., & Gołębiowski, P. (2014). Models of traffic flow distribution for various scenarios of the development of proecological transport system. Archives of Transport, 32(4), 17–28.

Kisilowski J., & Zalewski J. (2021). An example of a power-off maneuver of a vehicle without a straight line motion control. Archives of Transport, 58(2), 63-80.

Kobryń, A. (2014). New solutions for general transition curves. Journal of Surveying Engineering, 140(1), 12-21. doi: 10.1061/(ASCE)SU.1943-5428.0000113

Koc, W. (2019). New transition curve adapted to railway operational requirements. Journal of Surveying Engineering, 145(3), 04019009.

Kufver B. (2000). Optimisation of horizontal alignments for railway – procedure involving evaluation of dynamic vehicle response. Linköping: Swedish National Road and Transport Research Institute.

Kukulski, J., Jacyna, M., & Gołębiowski, P. (2019). Finite Element Method in Assessing Strength Properties of a Railway Surface and Its Elements. Symmetry-Basel, 8(11), 1–29.

Kukulski, J., Gołębiowski, P., Makowski, J., Jacyna-Gołda, I., & Żak, J. (2021). Effective Method for Diagnosing Continuous Welded Track Condition Based on Experimental Research. Energies, 14(10), 2889. doi:10.3390/en1410288

Li, X., Li, M., Wang, H., Bu, J., & Chen, M. (2010). Simulation on Dynamic Behavior of Railway Transition Curves. In ICCTP 2010: Integrated Transportation Systems: Green, Intelligent, Reliable (pp. 3349-3357).

Long, X. Y., Wei, Q. C., & Zheng, F. Y. (2010). Dynamic analysis of railway transition curves. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 224(1), 1-14.

Makarova, I., Shubenkova, K., & Pashkevich, A. (2021). Efficiency assessment of measures to increase sustainability of the transport system. Transport, 36(2), 123-133.

Pielecha, I. (2021). Energy management system of the hybrid ultracapacitor-battery electric drive vehicles. Archives of Transport, 58(2), 47-62.

Pirti, A., Yücel, M. A., & Ocalan, T. (2016). Transrapid and the transition curve as sinusoid. Tehnicki vjesnik, 23(1), 315-320.

Shen, T. I., Chang, C. H., Chang, K. Y., & Lu, C. C. (2013). A numerical study of cubic parabolas on railway transition curves. Journal of Marine Science and Technology, 21(2), 11.

Szaciłło L, Jacyna M, Szczepański E, & Izdebski M. (2021). Risk assessment for rail freight transport operations. Eksploatacja i Niezawodność – Maintenance and Reliability, 23(3), 476–488. doi: 10.17531/ein.2021.3.8.

Tari, E., & Baykal, O. (2005). A new transition curve with enhanced properties. Canadian Jour-nal of Civil Engineering, 32(5), 913-923.

Urbaniak, M., Kardas-Cinal, E. & Jacyna. M. (2019). Optimization of Energetic Train Cooperation. Symmetry, 11(9), 1175. doi:10.3390/sym11091175

Xu, Y. L., Wang, Z. L., Li, G. Q., Chen, S., & Yang, Y. B. (2019). High-speed running maglev trains interacting with elastic transitional viaducts. Engineering Structures, 183, 562-578.

Zboiński, K. (1998). Dynamical investigation of railway vehicles on a curved track. European Journal of Mechanics-A/Solids, 17(6), 1001-1020.

Zboiński, K., & Woźnica, P. (2012). Optimisation of railway polynomial transition curves: a method and results. In Proceedings of the First International Conference on Railway Technology: Research, Development and Maintenance, Civil-Comp Press, Stirlingshire.

Zboinski, K., & Woznica, P. (2018). Combined use of dynamical simulation and optimisation to form railway transition curves. Vehicle System Dynamics, 56(9), 1394-1450.






Original articles

How to Cite

Zboiński, K., Woźnica, P., & Bolzhelarskyi, Y. (2021). Modelling of the shape of railway transition curves from the point of view of passenger comfort. Archives of Transport, 60(4), 205-217.


Similar Articles

1-10 of 233

You may also start an advanced similarity search for this article.