Probabilistic vibration-based weight-in-motion solution for ballasted tracks of railway networks

Exceeding permissible axle loads and uneven load distribution between axles are critical safety and maintenance concerns for railway networks. Most existing weigh-in-motion (WIM) systems rely on rail deformation measurements. This paper proposes a vibration-based WIM method for ballasted tracks, in which the acceleration time history measured at the midspan between two sleepers is used to estimate train speed and axle loads. A probabilistic, mechanics-based framework is developed to interpret rail vibration measurements. An approximate closed-form expression is derived to predict rail acceleration at the forcing frequency. Its predictive accuracy is improved by adding a probabilistic correction term, calibrated through a Bayesian approach using results from Monte Carlo simulations. These simulations model the dynamic response of the train–track–ballast system, with input parameters randomly sampled within realistic ranges. The vehicle is represented as a two-degree-of-freedom vertical–pitching system, and the track is modelled with discrete sleeper supports. The framework is validated against experimental measurements on a ballasted track section. The proposed method provides a fast and practical procedure for estimating both the mean value and uncertainty of axle loads, given axle spacing, sleeper spacing, rail properties, and the measured peak acceleration at the forcing frequency.