An analytical model to calculate the forced vertical vibrations of two rails subjected to the dynamic loads of ballasted railway track

The railway track consists of two rails which are solicited the loads from passing trains. Due to the dynamic nature of these loads, the total forces applied to each rail are unequal and asymmetric. However, analytical models that simplify the track as a single beam cannot accurately describe the track responses. In this paper, we calculate the forced vertical vibration of a ballasted railway track subjected to dynamic loading with dual rails. Each rail is modelled as a uniform Euler-Bernoulli beam of infinite length, positioned on a system of periodic and identical supports. The forced vibration of the rail is determined using Floquet's theorem. Additionally, a model of a beam on a continuous foundation is employed to describe the dynamic behaviour of each support. In the frequency domain, each support is represented as a spring with an equivalent stiffness. By combining these two developed models, we analytically obtain the forced vibrations of two rails in the frequency domain. In the symmetric configuration, our results show a similarity of the rail responses with an existing model, particularly at the pin-pin resonance. The differing peak resonances at low-band frequencies are attributed to the different types of support. Subsequently, we apply this new model to investigate the forced vibrations of two rails in a non-symmetric configuration. The numerical tool visually illustrates the influence of load positions on rail responses and reaction forces in the time domain. This analytical model proves valuable for studying the rail noise and vibration of ballasted railway tracks.