Vertical vibrations of a freight car as a uniaxial carriage from a wave of unevenness of the path

Djabbarov, Shukhrat; Saidivaliev, Shukhrat; Abdullaev, Bakhrom; Inagamov, Sardor; Abdusaid, Yuldashov · 2023 · Crossref

DOI: 10.1051/e3sconf/202337104035

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Summary

This paper addresses the mathematical modeling of vertical vibrations in a loaded freight car, specifically treating the vehicle as a uniaxial carriage subjected to a wave of path unevenness. The research is motivated by the limitations of previous studies that modeled oscillations using piecewise linear problems with dry friction, which complicated the rigorous analysis of forced oscillations. To simplify this, the authors employ an approach based on S. P. Timoshenko’s method, replacing the constant dry friction force with an equivalent viscous drag force. This substitution is justified by equating the energy dissipated per cycle in both systems, allowing for the derivation of a linear inhomogeneous differential equation with constant coefficients. The methodology involves deriving the equation of motion for the relative vertical displacement of the wagon body. The authors calculate the energy dissipated by dry friction ($A = 4BF_{fr}$) and equate it to the energy dissipated by viscous resistance ($E = \pi b_0 B^2 p$) to determine the equivalent viscous resistance coefficient ($b_0$). Using Newton’s second law for relative motion, they formulate a second-order differential equation describing the system’s dynamics under a harmonic disturbing force caused by track irregularities. The solution to this equation yields expressions for the steady-state forced oscillation amplitude ($B$) and phase angle ($\gamma$), accounting for the system’s mass, spring stiffness, and the equivalent damping coefficient. The results demonstrate that the motion consists of transient free oscillations, which attenuate rapidly, and steady-state forced oscillations that persist. The study derives a specific formula for the amplitude of forced vibrations, introducing a dynamic coefficient ($k_d$) that depends on the detuning coefficient (ratio of disturbing frequency to natural frequency) and the ratio of friction force to the maximum disturbing inertia force. The analysis reveals that the dynamic coefficient becomes infinite at resonance ($p = k$), indicating that resonant amplitudes tend toward infinity even with slight friction, as the energy supplied by the disturbing force exceeds the energy dissipated. Additionally, the phase angle is shown to be largely independent of the frequency ratio except at resonance, where it changes abruptly. The significance of this work lies in providing a simplified yet accurate analytical framework for predicting the vertical vibration parameters of freight cars. By converting dry friction into an equivalent viscous model, the authors enable a complete analysis of forced oscillations without the complexity of piecewise linear solutions. The derived formulas allow engineers to determine the resistance coefficient and predict vibration amplitudes based on track irregularities, aiding in the assessment of freight car stability and suspension performance under operational conditions.

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• road surface friction