RAILWAY TRACK REPRESENTATION IN MATHEMATICAL MODEL OF VEHICLES MOVEMENT

Dep. «Roads Design and Construction», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel +38 (056) 373 15 48, e-mail kunibor@gmail.com, ORCID 0000-0002-8182-7709 Dep. «Track and Track Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 42, e-mail kurhan.d@gmail.com, ORCID 0000-0002-9448-5269


Introduction
The tasks of modeling the interaction of track and rolling stock are the basic ones for most areas of modern scientific railway transport research. Depending on the task being solved, one can use both relatively simple, sometimes even flat calculation schemes and the developed models de-scribed by systems with dozens of equations. Despite the fact that it is always being modeled the interaction process between track and rolling stock, the tasks of rolling stock research and those aimed at the railway track research have fundamental differences.
Today one can not recommend a single mathematical model and, even, a single modeling 40 principle to solve the entire variety of tasks of interaction between the track and rolling stock. It is the problem definition should determine the permissible hypotheses and assumptions, obligatory factors of influence, possible accuracy of the output data and sufficient accuracy of the obtained results and other characteristics determining the choice of the adequate for this case model.
It is rather difficult to obtain satisfactory results trying to combine simulation of rolling stock and railway track operation. Therefore, the most commonly used are not general models, but a staged modeling of processes in several different models. Thus, in the models of rolling stock with detailed accounting of dynamic processes in a vehicle and maximal simplification of rail track, the forces acting on the track are obtained. Further, such forces become the initial load (usually a simplified one to the static application of these forces) in models with detalization of rail track operation. This approach to the consistent application of two fundamentally different models is becoming increasingly widespread today.
Models of rolling stock are, in most cases, systems of motion (oscillations) of a set of interconnected solids. Usually, for the mathematical description of such models, the systems of differential equations of the second kind, compiled according to Lagrange-d'Alamber principle are used [1,2].
As a rule, it is taken a constant mass of interacting bodies, linearity of the connections between them, constant speed of motion, immutability of geometry of the bodies themselves (the deformations of system are determined by the change in position of bodies, not by their own deformation), the invariance of the physical (especially the elastic ones) characteristics of bodies and so on. Attempts to eliminate such assumptions considerably complicate solution of the equation system, and in most cases, it makes impossible to obtain the solution with the necessary accuracy at all. Particular attention should be paid to the fact that the system of bodies, described by the Lagrange equations of the second kind may have a local or global coordinate system. From the point of view of mathematical tools, this does not have a fundamental difference, but it has significant disadvantages for practical application. The local coordinate system assumes that its center moves together with the system, as a rule it is located in the center of the weight of the largest body, for example, the car body. Then the coordinates of all bodies at each step of calculation are moves relative to this point. At each step of calculation, they have values about one order, both between themselves and between values at other steps. This approach provides a certain accuracy of results of solving the equation system at each step in each direction. However, with this approach, a vehicle under study does not move along the track, but only fluctuates in one place. Local track inequalities are set in the same local coordinate system, and the global change in the motion trajectory is taken into account as application of the corresponding external forces. Thus, for example, the motion in a curve is set by applying to the bodies of the corresponding centrifugal force.
Global coordinate system establishes a permanent point of its center, the position of which does not change during the entire sequence of calculations. It simplifies the mechanism for describing the track position and makes it possible to set its spatial position as an array of coordinates. The advantages of this approach are, first, the ability to set any layout of the track, including the results of outdoor shooting; and secondly, the forces arising from changing the motion trajectory will be calculated as a consequence, not set as the source data. However, all displacements of individual bodies of car will also be determined relative to the originally set coordinate center. This will increase the calculation error.
Consequently, for the tasks in which it is the rolling stock dynamics that is under study, the local coordinate system is used -this is a more common variant. In some cases, when it is important to study the motion dynamics, depending exactly on the track geometry the global coordinate system is used.
Modeling according to the principle of Lagrange-d'Alamber has found a very broad application for solving the tasks of rolling stock dynamics. In accordance with the principles of solid state mechanics it is assumed that the object has a constant mass. The force applied to the object point instantly leads to displacement of all its other points and the body moves as one whole relative to the mass center. The use of such an approach for modeling the operation of a railway track construction may take place, but for the most tasks it should be considered inappropriate. One can divide a railway track into integral objects with a constant mass on-національного університету залізничного транспорту, 2017, № 6 (72) ly conditionally. Moreover, these masses will have small displacement values, which will occur in a short period of time.

Purpose
The work is aimed to create a methodology for representing a rail track in mathematical models of interaction with rolling stock and obtain practical results for various characteristics and track design and the maximum speed level.

Methodology
Rail track representation in the model of vehicle movement can be carried out in several ways, which, first of all, will differ in different detalization.
One of the variants involves description of rail track as an endless beam, which lies on an elastic, non-inertial basisthe Winkler model. This model is the basis of many static calculation schemes and is adequate for many tasks. But such a system will have an infinite number of freedom degrees, which is inconvenient for inclusion in rolling stock models. An alternative is the Vlasov model, which makes it possible to express displacement of the beam (rail) points and underrail base through displacement of the contact points of wheels and rails.
In this case, the general approach is maintained: a system assembly of separate rigid bodies with mass and connections between them. In the simplest form it can be a tight connection of wheel with a rail track, Fig. 1, a. Then the stiffness of such a bearing reduced to one wheel can be determined by the formula where Uis an elasticity modulus of underrail base; k -relative stiffness coefficient. This variant is rarely used. The absence of a dissipative connection makes the system too sensitive to the swing: the accumulation of errors in the solution of equations can lead to continuous oscillations.
In most cases, it is used the variant where each wheel has a rigidly dissipative connection with the reduced part of the rail track - Fig. 1, b. Today, it is spreading the tendency to take into account the track inertia, that is, a track (or its individual elements) should have a mass, which makes it an integral object of the mass oscillation system and makes it possible to detalize for separate components. The weight implies the so-called «reduced mass»that is, the one interacting with the wheel. Sometimes it means the mass of rail, sometimesthe mass of the track upper structure (again, reduced to one wheel). It is clear that in any case this mass is conditional, its value should change in the process of interaction, but systems based on the Lagrange-d'Alamber princile make it impossible to use variable masses. In addition, the railway track operates in the conditions of elastic deformations that can not be fully identified by the displacement of mass centers of rigid bodies.
Depending on the detalization degree of subsystem «rail track», the following variants are possible: the reduced mass of rail is taken into account; it is assumed that the rail and the wheel, as an object with their total mass, have a rigidly dissipative connection with underrail base, Fig. 1, c; it is taken into account the reduced mass of rail, which has a rigidly dissipative connection with underrail base; to separate the wheel mass from the rail between them a rigid connection, (with a large numerical value) is established, Fig. 1, d; it is taken into account the reduced mass of the track is taken into account, which has a rigidly dissipative connection with the base, Fig. 1, e; it is taken into account the weight of sleeper (interacting with the wheel), which has a rigidly dissipative connection with under sleeper base on the one side and with a rail on the other (as a rule, unimportant one (Fig. 1, f), although there may be the variants of rail mass separation too (Fig. 1, g); it is taken into account the weight of sleeper (interacting with the wheel), which has a rigidly dissipative connection with the rail on the one side and with ballast on the other, which has a rigid connection with sub-ballast base (Fig. 1, f).
From the point of view of vehicle movement modeling the variants from "f" (see Fig. 1) and further are not specify the calculation results as they complicate the general system of equations and can be identically referred to the variant "e" or even simpler ones. Such detalization takes place when trying to simulate the railroad operation itself. національного університету залізничного транспорту, 2017, № 6 (72) Thus, the task of presenting a rail track in models of rolling stock is reduced to determining characteristics of connections between its elements and the wheel. They can be obtained either by purely analytical approaches, or according to the results of dependences of the track deflections on the applied load. The latter variant can be implemented either experimentally according to the results of field measurements, or theoretically based on the results of variant calculations using corresponding models of the stress-strain behavior of the rail track. A similar problem was solved by Prof. O. M. Darenskyi in the work [3]. This work based on analytical calculations determines the support rigidity reduced to one wheel, for the zone of rail joints in the conditions of industrial transport operation.
In order to solve such problem for mainline transport, especially for the conditions of passenger traffic, which can occur at high speed [4], one of the important characteristics of the rail track model which can be applied is the ability to take into account the dynamics of the underrail base deflection. Thus, the task consists of determining such characteristics of the track as the reduced mass, rigidity coefficient, and dissipation coefficient. To solve the task let us consider the following calculation scheme, Fig. 2. Operation of the track reduced to one wheel is considered as a system with one freedom degree consisting of mass that has a rigid ( t k ) and dissipative ( t b ) connection with the base. An external force (P) variable in time is applied to the system. Such calculation scheme is identical to the rail track representation in the Lagrange-d'Alamber models shown in the Fig. 1, e, but, if necessary, it can be reduced to the other variants.
Differential equation for describing fluctuations of such system will have the following form If one integrates along the length of the covered distance, then where Vis the movement speed. The equations (3) are reduced to the classical form ; The force will be presented using the expression where Ais the parameter determining the force amplitude.
Then solution of the equation (4)  Let's assume that the wheel covers the distance from one axle between sleepers to the other, Fig. 3, and then the process is cyclically repeated. This approach corresponds both to the models in which the vehicle (wheel) movement along the track is conditional (wheel position relative to the track does not change in the local coordinate system, and movement relative to the track is taken into account by applying the corresponding external forces (accelerations, constraints), and the models in which movement along the track is set in explicit form, taking into account the wheel position along the track length. As a tool for solving such a problem it was used a model of stress-strain behavior of the rail track based on combination of equations of elastic wave propagation to describe the geometry of the outline of the part of the system space that is involved in the interaction at a given time and the equations of dynamic equilibrium of its deformation [5].