METHOD OF CONSTRUCTING THE DYNAMIC MODEL OF MOVEMENT OF THE MULTI-MASS SYSTEM

Dep. «Cars and Cars Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 04, e-mail reidemeister@mail.ru, ORCID 0000-0001-7490-7180 Dep. «Cars and Cars Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. La-zaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 793 19 16, e-mail kv47@i.ua, ORCID 0000-0002-8073-4631 Dep. «Cars and Cars Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. La-zaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 04, e-mail tri_s@ua.fm, ORCID 0000-0002-8256-2634


Introduction
At present, the approach to constructing models of car oscillations is well known and tested for assessing the car running qualities and dynamic loading of elements [12]. The car is considered as a set of solid bodies [1], connected by rigid, elastic, viscous, dissipative elements [10,11,13]. As a rule, the angles of rotation are considered small values, after which the compilation of the equations of motion becomes a routine procedure, which it is advisable to deliver into charge of the computer.
Let us consider one of the possible approaches to the solution of this problem, for this we will present a semi-formal way of describing the model and the rules of compiling the equations of motion. Actually, we will not be interested in the solution of the equations, since for these purposes there are universal packages of applied programs Simulink [6], Simscape, OpenModelica, Dymola (the last two packages implement the language of the description of dynamical systems Modelica [9]) and so on.

Purpose
To develop a methodology for describing the structure of the railway vehicles (they are considered as a system of rigid bodies connected by rigid, elastic and dissipative elements), which would allow us to obtain the equations of motion in an easily algorithmized way.

Methodology
When building a model, we strive to ensure that its structure reflects the structure of the mechanical system (car), that is, parts of the model must correspond to parts of the car. In this case, the model takes the form of a hierarchically organized graph whose vertices correspond to the bodies and connecting elements, and the edges describe the sets of processes that that are related to the incident to edge vertices [7]. An example of the general structure of the model is shown in Figure 1. As a rule, a set of generalized displacements and corresponding generalized forces correspond to the edge.
For the edge, causality conditions can be defined (for example, if the force is considered as a function of displacements) or not (forces and displacements are related by implicit relations). The difference between the two types of edges is not fundamental and, if desired, one can write down the formal rules for determining the causality relation.
To describe the model-building rules, we use the inductive approach and consider the basic types of subsystems and the corresponding equations. In doing so, we will try to match the set of equations to the node, and the set of variables to the edge.  Figure 2 shows a fragment of the «Bogie» subsystem, in which the bolster NB is connected to the solebars BR1, BR2 with the spring suspension unit RP1, RP2. The motion of such a subsystem is described by the equations that can be conveniently divided into the following groups: 1) Equations of motion of bodies; 2) Equations expressing the movement of the attachment points of the connecting elements through the movements of the bodies; 3) Equation of the relationship between the deformation of the connecting element and the force that arises in it.
The last group of equations refers to the connecting elements, the first two to the bodies. In view of the fact that the parameters of the equations of the first two groups are different, it is advisable to equip each body with an internal structure, as shown in Fig. 3, using the example of bolster. The inner vertex «Body» corresponds to the body motion equations. Internal vertices RP1, RP2, PP, SK1, SK2 -to the attachment points of corresponding connecting elements: spring suspension, pair «Centre plate -Centre pad», side bearing. These classes of vertices (for the body, for attachment points of the connecting element and for the connecting element itself) are the basic ones for building the car model. We will dwell in detail on each of them.
The vertex representing the motion equations of the body, whose principal central axes of inertia are parallel to the coordinate axes, is shown in Fig. 4.
This vertex can be incident with several edges, each of which is associated with a set of generalized displacements ( ) j q and generalized forces where m -body mass, , , Variables corresponding to displacements are called variables of the potential type, and processes corresponding to the forces are variables of the current type. These names refer to Kirchhoff's laws for electrical circuits and to the fact that the movements in the edges incident to one vertex are equated to each other, and the forces are added together.
, q Q take the form:  The «Connecting element» connects the «Points» of two «Bodies», Figure 6. The deformation of the connecting element is the difference ( The force Q , arising in the connecting element depends on the deformation q  . The expression for the force depends on the type of the connecting element. For example, for a linear spring of rigidity C , operating in a vertical direction 0, , 0.
It is convenient to assume that the force Q acts on the Body-0 from the side of the Body-1. In this By combining the described types of vertices, it is possible to present in a compact and visual form a model of car oscillations, suitable for direct formation of the motion equations.
Let us consider the implementation of the described approach for building a model in the Simulink package. In this case, the body, point and connecting element are conveniently represented as subsystems. As an example, Figure 7 shows the Simulink scheme for the subsystem «Bolster». Having examined the expression (2), we see that the directions for displacements and the directions for forces in one edge are opposite to each other. The direction of propagation for different signals is given in Table 1. Table 1 Directions of signal propagation

Vertex class
Process Direction

Findings
The use of the proposed method resulted in creation of a freight car model, which consists of a body and two bogies. The bogies were considered as a construction consisting of the following subsystem-elements [2,3]: wheel sets with box; -solebars; -bolster; -springs of the central spring unit (each tworow spring separately); -friction vibration dampers; -axle-box suspension; -centre pad; -side bearing. Each of these subsystems is independent and can be replaced with the condition of preserving the geometric parameters of the connection points of the element. This feature of the model is convenient to use for changing the parameters in order to select their optimal values.
The bodies, with the exception of bolsters and bogie solebars, have 6 degrees of freedom; the angles of rotation are small. To better estimate the dynamic loading of the bogie cast parts, the detailed models of spring suspension (up to individual springs and friction wedges, which are considered as separate bodies of the system) and axle box were developed.
The model will be used to evaluate the dynamic loading of bogie elements of the car with an axial load of 25 tons [8], in order to clarify the loads arising during operation [4,5,14]. Vertical forces acting in the axle box when the car moves along a straight section of the track at 120 km/h speed are shown in Figure 8.

Originality and practical value
For the first time, a methodical approach to creating dynamic models of railway vehicles based on their description using hierarchically organized graphs was proposed. This methodological approach will allow, after creating a library of bodies and connecting elements, to significantly reduce the time spent on modeling the oscillations of different vehicles.

Conclusions
A technique has been developed for describing the structure of a railway vehicle using a hierarchical graph, which makes it possible to obtain equations of motion in an easily algorithmized manner. The vehicle is a system of rigid bodies connected by rigid, elastic and dissipative elements. The technique was tested to construct a model of spatial oscillations of a 4-axle freight car in the Simulink package. Directions of further development: creation of the library of bodies and connecting elements, detailed presentation of the geometry of track, the models of track superstructure and wheel-rail interaction.