Optimization of Polynomial Transition Curves from the Viewpoint of Jerk Value

Abstract

The aim of the presented paper is to show the results of shape optimization of railway polynomial transition curves (TCs) of 5th, 7th, and 9th degrees through the use of the full vehicle model and new criteria of assessement concerning the jerk value. The search for the proper shape of TCs means that in this work, the evaluation of TC properties is based on select quantities and the generation of such a shape through the use of mathematically understood optimization methods. The studies presented have got a character of the numerical tests. For this work, advanced vehicle models describing dynamical track-vehicle and vehicle-passenger interactions as well as optimization methods were exploited. In the software vehicle model of a 2-axle freight car, the track discrete model, non-linear descriptions of wheel-rail contact are applied. This part of the software, the vehicle simulation software, is combined with a library optimization procedure into the final computer program.

Railway Transition Curves: A Review of the State-of-the-Art and Future Research

“…In the group that uses advanced vehicle-track dynamics the research is performed by Zboi ński together with Woźnica or Golofit-Stawinska. The two topics they focus on are forming of polynomial transition curves with a combined use of optimization methods and advanced vehicle models [43,50,51], and investigating the non-linear phenomena that occur in the motion of railway vehicles in transition curves with speeds close to the vehicle's critical velocity [58]. The authors identify a change in content of papers concerning transition curves where an attempt to get away from the standard approach of evaluating curve properties and finding new methods is made [51].…”

“…Based on this they observe a lack of research where advanced dynamics and optimization methods are exploited to investigate interesting phenomena of railway vehicles in transition curves and form optimal transition shapes. The curves chosen for optimization in [43,50,51] are polynomial curves of degree n ≥ 4, and the objective is to find the polynomial coefficients A i , that define the curve's shape, in the following expressions (curve, curvature, superelevation, inclination of superelevation ramp and velocity of wheel vertical rise along the curve)…”

“…The introduction of lateral change of acceleration (LCA) in 1996 [4] showed that with the increasing speeds in rail transport, the clothoid was not as optimal any more (more specifically for speeds above 120 km h [29]). This realisation opened the door to research on new types of transition curves, including parabolas [41], sinusoids [42], polynomial curves [43] and spline curves [44], which enjoy smoother LCA than the clothoid.…”

Smoothed transition curve for railways

“…We assume conducting analysis for train speed V = 100 km/h. Assuming the cant value on a circular arc h 0 = 70 mm and the allowable value of unbalanced acceleration a dop = 0,85 m/s 2 , based on condition (21) we get the value R min = 590,002 m. In the calculations that follow, we take the radius of the circular arc R = 600 m. Assuming the permissible value of the acceleration change speed ψ dop = 0,3 m/s 3 (as for single transition curves with linear curvature), based on conditions (23) and 24, we determine the minimum lengths of the respective transition curve and finally we assume: (15) and (18). Figure 4 shows the appropriate horizontal ordinates graphs based on formulas (13), (14), (16), (17), (19) and (20).…”

Analysis of the possibilities in railways shape assessing using GNSS mobile measurements

“…The relations concern especially the admissible train speed as well as the displacements and transverse accelerations occurring on the rail vehicle. On this basis, simulation models of vehicle and track interaction are still being developed [1,2], and also new methods for mathematical and numerical description of the track axis are proposed [3,4,5], as in case of road engineering -especially in case of highways design [6].…”