## 1 Introduction
Geometric defects of wheels and rails are rarely found at initial stages of usage; however, over time, these defects increase. In order to describe the interaction between the flat wheel and rail, many authors of mathematical models described the wheel and rail contact area as a point or the wheel geometry as an analytical function [15; 17; 18; 19; 20; 21; 22; 23; 24]. In order to simplify mathematical models and considering that the most common damage of the rolling stock wheels is a wheel with a flat spot, the authors usually choose the flat wheel damage [7; 25]. The interaction between the moving wheel and the rail is analyzed [2] taking into consideration the rail unevenness and the axial load of the rail, which affect the contact properties of the wheel and the rail. Moreover, the dynamic processes of the developed system are investigated. The simplified wheel and rail nonlinear Hertz contact model consists of two masses with elastic connection. The mathematical models of wheel-rail interaction can be divided into two groups: when the total area of the wheel and rail contact is calculated and when the contact area is halved. Two mathematical models [26] of wheel and rail interaction were created. The first mathematical model calculates the total area of the wheel and rail contact, while the second divides the contact area in half. The obtained results of wheel and rail contact were compared with each other and the significance of the elasticity and stiffness analysis for the contact stress testing was established. A mathematical model of wheel and rail interaction [27] was developed, where the wheel and rail are described by linear equation systems using the impulse reaction function. The model allows the calculation of vertical contact force caused by unevenness of the rails and wheels. The results of the research on acting forces [28] caused due to interaction of flat wheel with a rail were given. These forces are determined by the axial forces acting on different rails and the train speed. It was determined that the maximum obtained impact force between wheel and rail varies for different static wheel loads, and, when the speed is relatively high, the flat wheel loses contact with the rail. During the investigation of the dynamic processes occurring in wheel and rail contact, the researchers described the variations in wagon and rail mass acceleration and displacement within a time or track section [3; 28; 29; 30; 31]. It was determined that the maximum acceleration value is obtained in the case of a wheel with a flat spot. The 1520 mm gauge railway infrastructure of the Republic of Lithuania is designed according to and must comply with the agreed technical parameters and methodologies of the railway infrastructure operators of the Organization for Cooperation between Railways (OSJD). Therefore, the Russian scientific literature must be considered, which contains works that use reverse Laplace transformation to develop the mathematical model of the system [32; 33; 34; 35; 36; 37], and the mathematical model of the wheel with a flat surface is derived through frequency characteristics. Recently, new methodologies of the theoretical part [38] for determining the discarding criteria \(P_{derivative}\) of the wheel with flat surface were established, using the mathematical models given in the literature [32; 33; 34; 35; 36; 37] and the description of the derivation of the mathematical model of the wheel with a flat surface, checking the results using a physical experiment. The results of this paper include an analysis of the automatic rolling stock monitoring systems used to evaluate the condition of the wheel axles of the rolling stock, whereas methods and criteria for discarding them were compared; additionally, a methodology for updating the ATLAS-LG system software operation algorithm and setting the new discarding parameter \(P_{derivative}\) was described.