This paper presents a mathematical model used to obtain the vertical vibration of a ballasted railway track when a wheel is passing at a certain speed over a fixed location of the rail. The aim of this simulation is to compare calculated root-mean-square (RMS) values of the vertical vibration velocity with measured RMS values. This comparison is the basis for a proposed time domain methodology for detecting potential wheel flats or any other singular defect on the wheel rolling bands of metropolitan trains. In order to reach this goal, a wheel–rail contact model is proposed; this model is described by the track vertical impulse response and the vertical impulse response of the wheel with the primary suspension, both linked through a Hertz nonlinear stiffness. To solve the model for obtaining the wheel–rail contact force, a double convolution method is applied. Several kinds of wheel flats are analyzed, from theoretical round edged wheel flats to different real wheel profile irregularities. Afterward, the vertical vibration velocity at a fixed point on the rail is obtained using a variable kernel convolution method. Running different simulations for different wheel flats, a study of the vertical vibration attenuation along the rail is carried out. Finally, it is proceeded to obtain the temporary evolution of the RMS value for the rail vertical vibration velocity in order to be used as a reference for detecting wheel flats or any other defect. This last aspect will be presented in more detail in a second paper.

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