The author has developed a mathematical model to study spatial oscillations of a tetra axial flat car with long goods (the ends of goods exceed limits of frontal bars of carriage underframe for more than 400 mm) leaned on two elastic dissipative supports of a flat car. At the same time the free ends (consoles) of a cargo overhang the floor of two cars. Thus the mechanical system «flat car – long goods’ consists of 12 solids: long goods which are elastic in vertical plain, frame of a car which is also elastic in vertical plain, two over spring beams, four lateral beams and four wheel pairs of model 18–100 bogies.

There are some allowances that help to elaborate a mathematical model: gaps in pivot plate assemblies are not taken into consideration; lateral rolling motion of side frames of bogies is absent; wobbling, transversal drift of side frames of bogies, as well as wobbling of over spring beams and wheel pairs are equal; rail track is rigid in vertical plain and elastic in horizontal plain. In order to define fundamental function of bending of long goods in vertical plain, the author used differential equation of free motion of a rod with constant cross-section along its length when it is placed on elastic foundation. Three forms of oscillation are composed. The first and the second forms refer to oscillations of long goods as of solids (bouncing and rocking), the third one refers to oscillations of load as of elastic body.

In order to compose differential equations, describing spatial oscillations of a flat car with long goods, moving along straight and curve track, the author used Lagrange equation. The elastic features of the bogie of a flat car as well as elastic features of long goods are provided for in kinetic energy. Describing the supports for long goods the author takes into consideration elastic and viscous forces and the moment. Generalized forces are defined through the forces in lateral bearers during side drifting that influence over spring beams of bogies and through the forces that result from relative travel of over spring beams and side frames of bogies, as well as through the moments of dry friction during wobbling of the frame of a flat car, through the moment of lateral rocking of the frame and through the moment of edge bearing of center plate on thrust bearing of a bogie. The author also has taken into consideration the forces of interaction between the wheels and the rails, caused by elastic motion of wheels along the rails, as well as the forces caused by conicity of the surface of wheel rolling.

The study resulted in a system of 20 equations which describe spatial oscillations of the mechanical system «flat car – long goods’ at the moment when it moves along straight track and track curves (circular curves). While motion in curves is analyzed, the absolute coordinates are assumed, which are equal to the sum of coordinates in relative and translational motion, the uncompensated lateral accelerations been also considered. Mathematical model assumes clinch, vertical and horizontal harmonic irregularities accounting also for transportation lag of whhel pairs.