Using the Fourier-series approach to study interactions between moving wheels and a periodically supported rail

In this paper, the Fourier-series approach is employed to study wheel–rail interactions generated by a single, or multiple wheels moving at a constant speed along a railway track. This approach has been previously explored by other researchers and what is presented here is an improved version. In this approach, the track is represented by an infinitely long periodic structure with the period equal to the sleeper spacing and the vertical irregular profile (roughness) of the railhead is assumed to be periodic in the track direction with the period equal to the length of a number (integer), N, of sleeper bays. By assuming linear dynamics for the wheel/track system and for steady state, each wheel/rail force is a periodic function of time and can be expressed as a Fourier series. Fourier coefficients are then shown to be determined by solving, separately, N sets of linear algebraic equations. The coefficient matrix of each set of equations is independent of rail roughness and therefore this approach is particularly useful in modelling the generation and growth of rail roughness of short wavelengths. Excitation purely from the axle loads moving over the periodic track structure is realised by assuming a smooth railhead surface, and subsequently roughness equivalent to such an excitation is defined and evaluated. This equivalent roughness may, in addition to the actual rail roughness, be input into models in which the effect of moving axle loads has been excluded, so that the predictions from those models can be improved. Results are produced using the improved Fourier-series approach to investigate the effects of wheel speeds, roughness wavelengths and interactions between multiple wheels on wheel/rail contact forces.