Train-weight-in-motion identification measuring time-histories of rail strains

This paper deals with the identification of the weight of a train in motion, based on the measurement of the time-history of the response in terms of strains at the foot of the rail. The direct problem is initially addressed: the response of a rail modelled as a one-dimensional Euler-Bernoulli beam with constant properties, resting on a linear elastic foundation with viscous damping and subjected to a Dirac delta load travelling at constant speed is considered. For the model described, a closed-form expression of the solution can be obtained, which permits to investigate the sensitivity of the response to the main mechanical parameters. Analytical strains are compared to their experimental counterpart, showing their practical ability to describe the real phenomenon. As a second step, the inverse problem consisting in the identification of the loads for a given time-history of measured strains is addressed. The solution of the inverse problem is set up as a minimization problem whose objective function is based on the difference between experimental and model time-histories of strains. This inverse problem is nonlinear, and its solution can be pursued by the Newton method, which requires recursive application of a linearized expression for the evaluation of the optimal parameters. The Bayesian formulation enables to investigate identifiability of parameters and minimum number of measurements, and leads to conclude that the identification process must begin with an improving of the interpretative model. This model updating can be achieved by evaluating the model parameters, using the time-history of a train whose weight is known. After that, the actual identification of the loads can be performed. The procedure proposed is applied to experimental strains recorded at the foot of a rail on a stretch of line run by a locomotor moving at a low constant speed. The identified loads were in good agreement with the expected value, with errors smaller than 4%.