## 1 Introduction  
On the basis of a comprehensive analysis of the safety of railway traffic conditions, including the derailment of rolling stock, it is evident that traffic safety violations sometimes occur even with all indirect security parameters maintained. If the rolling stock and track interaction forces could be controlled directly, such accidents could be avoided. To ensure traffic safety and reduce the number of residual defects on the track, the impact of the infrastructure's rolling stock on the railway track must be assessed. This conclusion obligates the engineers to develop systems that assess not only the secondary, but also direct parameters that determine the traffic safety and the reduction of residual deformation and its accumulation in the structural elements of the rolling stock and the track. A serious achievement in this area is the development and implementation of the automated identification of rolling surface defects according to the values of interaction forces. Here, detection is based on detection of forces that act during the contact of the rolling surface with the rails. It is important to understand the influence of force acting between the rolling surface and rails, defining the contact force limits when the overload causes defects on the wheels. The main elements of the calculation scheme of the mathematical model of the system \"Rolling Stock-Track\" used by many authors [1; 2; 3] are the \"rolling stock\" (rolling stock-wheel)-\"contact between wheel and rail\"-\"track\" (rail-gap-railway sleeper-trackbed). The distribution of the forces generated at the wheel and rail contact point affects the dynamics of the wagon and the track. The contact forces of the defective wheel and a rail have a stronger effect than the healthy wheel and the rail. The magnitude of the contact forces depends directly on the speed of the wagon, geometrical parameters of the wheel and rail, and physical and mechanical characteristics of the rolling stock and track. The prediction of wheel and rail interaction is important not only for the operation of railways, but also for modeling the dynamics of multiple bodies. Attempts were made to improve dynamic models by implementing the basic factors related to wheel and rail coupling in the calculation of interaction models [4; 5]. Mathematical models of rails are used for various complexity theories. It is possible to distinguish the main mathematical models [6; 7; 8] that are most common among authors: [PERSON] and [PERSON]. The study of wheel and rail interaction is based on wheel and rail interaction theories: [PERSON] [9], [PERSON] [10], [PERSON] [11], and others. [PERSON] extensively described [PERSON] and [PERSON], as well as their capabilities, limitations, and adaptations [11]. [PERSON]'s theory describes the wheel's acceleration and braking action when large tangential forces are transferred to the rail. The rail is considered to be an infinite beam. When developing mathematical models of a moving wagon, [PERSON]'s method considers the longitudinal contact force of wheel-rail but does not consider the transverse forces of a wagon. The most common is the [PERSON] theory of elastic bodies [12; 13; 14], which states that, when two oval bodies are compressed, the contact area is elliptical, and the maximum pressure is generated at the center of the contact zone. By integrating the pressure distribution in the contact zone, the maximum load can be obtained. The wheel with a flat spot can cause a significant dynamic effect on the entire system under consideration. A larger wheel's flat spot results in a larger impact force. A mathematical model [15] was designed to investigate the vertical loads between a flat wheel and a rail. It was determined that the impact force increases with the increase in the size of the flat spot. A mathematical model of interaction between two rigid contacting bodies was developed to estimate the friction forces [16]. The interaction contact area is considered as a point plane.