An analytic solution of mathematical expectation for bogie-track interaction problems

Abstract

This paper presents a frequency-domain analytic solution of mathematical expectation for bogie-track dynamic interaction problems in which a random roughness on the railhead is considered. To achieve this, the Floquet transformation is applied to the coupling system consisting of a bogie and an infinite track. Due to the periodicity of the sleeper spacing, the present problem is reduced to that in a representative track unit. Transformed solutions are expressed by Fourier series in the unit cell. The unknown Fourier coefficients are then obtained from infinite simultaneous equations. In the present formulation the vibration reaction due to the roughness is described in terms of the response function of the bogie-track coupling system and the power spectrum density (PSD) of rail roughness. The track dynamic nature which is independent of the roughness is represented by the former. To validate the derived solution, comparison with time-domain numerical solution is carried out. Furthermore, based on the developed frequency-domain method, two kind of pad damping model given by a constant loss factor and viscous damping are compared through evaluation of the expected value of energy spectrum density (ESD) of rail vibration and the PSD of wheel acceleration. Finally, influence of track structure such as the dynamic equivalent stiffness of rail pad and that of under-sleeper pad on the expectation of ESD of rail vibration and the PSD of wheel acceleration is examined.