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It is shown that dynamic systems, «rolling stock - way» due to the unevenness of the path on length should be described by ordinary differential equations with variable coefficients, the method of analyzing differential equations with constant, variable and random coefficients describing the movement of electric locomotive nodes when they move along an uneven path. In the transition to a new paradigm, we can talk about areas of dynamic instability, which in the case of simple parametric resonances develop near critical frequencies, but this is not one specific point, but a zone that expands with increasing coefficients of parametric excitation. In addition, the presence of friction in the system does not guarantee the limitation of resonant amplitudes. The effect of parametric arousal factors on the width of the dynamic instability zone has been established. There are many other features in the behavior of differential equations with variable coefficients, so it is impossible to replace the action of unevenness with some equivalent geometric irregularity, since at this moment there is no exact solution to the problem with which to compare the results of approximate mathematical models.

Method of construction of mathematical models is offered for mechanical systems with additional feedback ties which formed of lever mechanisms of different kinds. It shown that mathematical models can be get by simplifying more complicated system in which lever ties are realized of rigid body with fixed pivot point. Lever ties at the preparation of a model formally appear at decrease inertial moment of intermediate rigid body to very small values. It shown that mechanical oscillation system can be reduced to equivalent scheme which matches estimation scheme of chain type. Equivalent estimation schemes can to differ of various kinds of between-partial ties. Possibilities of using quasi-springs in equivalent estimation schemes are offered and justified. These complicated compositions consist of elastical elements which interconnected of lever mechanisms. Coerced stiffness’s of quasi-springs defined on rules of transformation of structural mathematical models. Base of mathematical modeling is using method of transformation of Laplas with following construction of equivalent in dynamical attitude of structural schemes of automation control systems. Analytical ratios are given. They characterize possibilities of special dynamical regimes.

The influence of parameters of the spring suspension of freight wagon and axial load on its dynamic qualities and traffic safety. Showing the flaws of the standard three-piece bogie, influencing the dynamics and traffic safety. Investigated the horizontal dynamics of the of freight vechicle and the value of horizontal stiffness of spring suspension bogie for unoaden vechicle .

Considered forced oscillations of four-axle vehicle with dual spring suspension. It is obtained that the movement system with six degrees of freedom with sufficient accuracy can be represented by a system with two degrees of freedom. Therefore, the vehicle body has two degrees of freedom: lateral skidding and wobbling, Bouncing and galloping trucks will be neglected. The total number of degrees of freedom of the model equal to two. The method, which leads to a closed solution of the forced oscillation. Defined. That tends and to (n = 1, 2,…), the amplitude of the oscillations will be unlimited. Therefore, the parameters k1, k2 and T must be selected on the basis of required values of amplitudes. As a result of research compiled by the differential equation of motion of the system and obtained analytical solution for the case of external piecewise constant driving force.

Currently, in the brake system of operated cars, there is a problem of lack of brake pressure on the wheel, which is necessary for the movement of the train at a set speed in accordance with the regulatory documentation. This is due to the tendency to increase the mass of trains and the maximum load of cars with a constant tare weight, which requires an extended range of pressure control in the brake cylinder by an air distributor or auto mode, depending on the load of the car. To solve this problem, calculations were made of the braking coefficient of cars equipped with air distributors № 483, which showed that the freight car was not equipped with brakes to run as part of a train at a set speed. In the course of engineering studies, it turned out that increasing the spring rate of loaded and medium modes can increase the adjusting range of the air distributor and partially solve the problem of lack of brake pressure. Thus, it is proposed to recalculate the stiffness of the adjusting spring. Calculations of the rigidity of regular and proposed springs were made. The braking coefficient has been recalculated, taking into account the increased spring constant. As a result, this coefficient has increased, which makes it possible to remove speed limits. The proposed method for upgrading the air distributor can be carried out on all types of repairs, which facilitates the implementation process.

In this paper the authors propose to consider the peculiarities of making mathematical models of rolling stock and its dynamic behavior when driving on irregular resilience railway line in the longitudinal direction. We present a qualitative and empirical analyzes of longitudinal resilience railway railway. In conclusion, given the conclusion, on the basis of its proposed further consideration of mathematical aspects of the above solutions of systems of differential equations of motion of the rolling stock on irregular resilience way.