Advances in Civil Engineering

Volume 2016 (2016), Article ID 6503962, 9 pages

http://dx.doi.org/10.1155/2016/6503962

## Design of Reverse Curves Adapted to the Satellite Measurements

Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland

Received 26 November 2015; Revised 25 February 2016; Accepted 13 March 2016

Academic Editor: Samer Madanat

Copyright © 2016 Wladyslaw Koc. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents a new method for designing railway route in the direction change area adapted to the Mobile Satellite Measurements technique. The method may be particularly useful in the situations when both tangents cannot be connected in an elementary way using a circular arc with transition curves. Thus, the only solution would be the application of two circular arcs of opposite curvature signs, that is, the use of an inverse curve. It has been assumed that the design of the geometrical layout will take place within an adequate local coordinate system. The solution of the design problem takes advantage of a mathematical notation and concentrates on the determination of universal equations describing the entire geometrical layout. This is a sequential operation involving successive parts of the mentioned layout. This universal algorithm can be easily applied to the computer software which will allow generating, in an automatic way, other geometrical layouts. Then, the choice of the most beneficial variant from the point of obtained trains velocities while minimizing the track axis offsets will be held using the optimization techniques. The current designing methods do not provide such opportunities. The presented method has been illustrated by appropriate calculation examples.

#### 1. Introduction

An inspiration for undertaking the analyzed problem is undoubtedly the new technology—the application to the railway track satellite measurements GPS. The global positioning system GPS [1–5] enables determining the coordinates of points in a uniform three-dimensional reference system WGS 84 whose origin is placed in the centre of the Earth’s mass. Effective measurements of railway track might be obtained by the method of Mobile Satellite Measurements elaborated by a scientific team of the Gdańsk University of Technology and the Naval Academy in Gdynia [6, 7]. This method refers to a pilot study [8] and involves driving through the tested section of the route being inspected by the use of antennas installed on a travelling rail carriage.

The Mobile Satellite Measurements make it possible to determine the coordinates of the existing railway route using the Cartesian coordinate system (which in Poland corresponds to the national spatial reference system [9]) as described in the works [10–13]. It would be advantageous if the newly designed track axis coordinates were determined in mentioned system; in particular, the coordinates are used in setting out the route in terrain.

General principles for design of track geometry were formulated in the nineteenth century. Then their continuous modification followed, which was reflected in the constantly changing regulations. It should be noted, however, that the development of computer technology constantly stimulates the ongoing search for new solutions, presented among others in [14–17].

The significant precision obtained in the method of Mobile Satellite Measurements in terms of the determination of coordinates in horizontal plane (with an error in the range of several millimeters) [12] inclines to work out new design methods adapted to the satellite measurement technique [18, 19] and to new computer-aided programs [20].

The interest of analyzed problem is confirmed by the investigations carried out in Europe in 2006–2010 as part of the INNOTRACK program [21]. The investigations were coordinated by International Union of Railways (UIC). Over 30 participants were engaged in the project including 8 leading infrastructure managing directors (among others from UK, Germany, and France). It appears that on the list of the most frequent problems raised by the infrastructure board of directors is the ascertainment “*bad track geometry.*” The methods of geometric shaping of tracks that have been used so far proved to be ineffective. Thus, there is plenty of work to do to improve the present unfavorable situation.

The difficulties connected with the geometric track shaping in horizontal plane result from the fact that the applied geometric elements, like straight sections, radii of circular arcs, and transition curves, are very often characterized by large lengths and therefore a visual evaluation of the whole system using the traditional geodesic techniques becomes ineffective. The system should be divided into parts and analyzed individually, which causes extra errors.

At this point it is essential to note a very important fact. The reason is that the railway project has its own characteristics, which is in fact generally connected with existing layout and the regulation of the track axis. In situations relating to regions requiring an alternative route direction, the design will, in principle, be based on making a correction of the circular arc radius as well as the type and length of the transition curves so that the new geometric layout is most desirable from the rail vehicles kinematic point of view. Simultaneously, its position in horizontal plane will not divert too much from the existing layout.

Determining the values of geometrical parameters which will ensure meeting these conditions becomes in fact a key issue. Specifying these parameters requires consideration of multiple design options and making appropriate choice, using optimization techniques. The new method of calculating the coordinates adapted to the satellite measurements is essential to generate variants of railway geometrical layout. Only after obtaining the appropriate values of the geometric parameters as a result of this procedure it is possible to use in a rational way any of the commercial computer programs supporting the designing process.

As proved by the satellite measurements that have been carried out so far, the shape of the railway track in operation can be so deformed that the determination of the main directions turns out to be impossible; one cannot apply a model system to the design: transition curve-circular arc-transition curve. The only solution in this case is to introduce two circular arcs of different radius to the geometrical layout, which means applying a compound curve [19].

All this, however, does not cover the whole problem yet. But under some conditions even the use of a compound curve does not allow us to connect the major directions of the route. In consequence, it is necessary to use two inverse arcs in the geometrical layout. A description of the design procedure for this type of situation can be found in the content of the paper. The presented conception of the designing technique related to a route realignment area creates an opportunity, like other elaborated methods, to obtain an analytical solution by the application of adequate mathematical formulae, which is more convenient for practical usage.

#### 2. General Assumptions

The route major directions in the Polish National Spatial Reference System can be defined by the following equations [9]:

Straight :

Straight :

In the above equations, and are absolute terms in the expression and and define the slope coefficients of the two straights. The straights have similar slope coefficient values and intersect at a distant point (they can also run in parallel to each other). Such a situation justifies the connection of both straights by reverse curve.

The designing procedure will take place within an appropriate local system of coordinates (LCS), making it possible to present the course of the route in functional notation. This system results from the adoption of the coordinates of its starting point on Straight 1 and making a rotation of angle. The problem of obtaining appropriate inclinations of Straights 1 and 2 in LCS system becomes a crucial question in this matter. The inclinations must be positive to operate the positive values of ordinates and also advantageous in the view of the procedure of determining these ordinates, that is, neither too big nor too small. In this respect the rotation angle will play a decisive role in it.

An assumption is made that, within system , , Straight 1 will pass through the centre of the system with a slope angle being equal to . For the reason that Straight 2 inclination value is similar to Straight 1 it will certainly be placed within interval , closer to the centre of that (i.e., inclination of Straight 1) than to its boundaries. A positive inclination of both straights ensures an analytical description of the whole geometrical layout by the use of explicit functions related to the circular arcs.

In order to insert an inverse curve between both straights it is necessary to find such coordinates of point , where Straight 2 can be placed on the right side of Straight 1. Knowledge of the point coordinates and the rotation angle enables mutual points transformation between global and local coordinate system. The value of turning angle is determined by the use of the following equation:where for and for . To obtain a positive value of angle from the above formula, the left turn of the system should be made, whereas, in the case of a negative magnitude, a right turn of the system should be made.

In assuming the coordinates of point along Straight 1 and determining the turning angle , it is possible to make a transformation of Straights 1 and 2 to the local coordinate system , . The entire geometric system in LCS system under consideration is presented in Figure 1.