Mathematical model of wagon wheels rolling along the hump profile

Djabbarov, Shukhrat; Saidivaliev, Shukhrat; Abdullaev, Bakhrom; Ortiqov, Mironshox; Rustamov, Nassim · 2023 · Crossref

DOI: 10.1051/e3sconf/202346006007

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Summary

This paper addresses the theoretical mechanics of railway wagon wheelsets moving along the hump profile of a marshalling yard, specifically focusing on the conditions for rolling versus sliding. The authors aim to clarify the causes of rolling friction moments and identify the precise conditions under which wheels slip rather than roll in braking zones. The study is motivated by the need to correct erroneous calculations of wagon sliding speeds, which often rely on simplified free-fall formulas that fail to account for the inertia of rotating parts and the complex friction dynamics between wheels and rails. The researchers developed a mathematical model based on the principles of geometric statics and theoretical mechanics. They formulated differential equations for the plane-parallel motion of solid body wheelsets, decomposing the movement into translational motion of the center of inertia and rotational motion around that center. The model incorporates various resistance forces, including gravity projections, air resistance, and curve resistance, while specifically accounting for the moment of rolling friction ($M_k$) arising from the local deformation of wheel and rail contact surfaces. Key parameters include the coefficient of sliding friction ($f$), the coefficient of rolling resistance ($f_k$), and the moment of inertia of the wheelsets. The analysis distinguishes between ideal connections (frictionless) and non-ideal connections (with friction and deformation). The results demonstrate that pure rolling without slipping is only possible under specific friction conditions. The authors derived a generalized formula for the linear acceleration of the wagon’s center of mass, showing that when wheels roll without slipping, the acceleration is significantly lower than that of a body in free fall due to the energy required to rotate the wheelsets. Specifically, the acceleration is approximately four times less than the projection of gravity on an ideal inclined plane when rolling friction is negligible. The study establishes a critical inequality involving the track inclination angle ($\psi$) and the friction coefficient ($f$) that determines whether pure rolling or rolling with slip occurs. If the friction coefficient is insufficient relative to the slope and inertial properties, the wheels will slip. The authors prove that using standard free-fall formulas for speed determination in braking zones is fundamentally flawed because it ignores the rotational inertia and the specific friction thresholds required for rolling. The significance of this work lies in its correction of existing theoretical provisions used in hump yard operations. By providing rigorous analytical formulas, the paper highlights that the determination of sliding speeds in braking positions must account for the transition between rolling and slipping states. This has practical implications for the safety and efficiency of railway marshalling yards, as accurate modeling of wheel-rail interaction is essential for designing effective braking systems and predicting wagon behavior on slopes. The findings underscore the necessity of considering rolling friction moments and rotational inertia in engineering calculations to avoid erroneous assessments of wagon dynamics.

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• road surface friction