Enhanced Formula for a Critical Velocity of a Uniformly Moving Load

The response of rails to moving loads is of interest in the area of high-speed railway transport. For determination of critical velocity of the train a theoretical concept that is based on the assumption that the track structure acts as a continuously supported beam (the rail) resting on a uniform layer of springs is traditionally used. In this contribution dynamic equilibrium of the soil in the vertical direction is implemented to obtain two frequency dependent parameters that are capable of handling geometric damping and of accounting for the soil mass inertia activated by induced vibrations. The new approach is tested on finite beams and single moving force. It allows for determination of resonant velocities. Then the quasi-stationary deflection shape of an infinite beam can be determined from two semi-infinite beams and critical velocities can be obtained from the nullity condition of the determinant of the dynamic stiffness matrix of the structure. 1 INTRODUCTION The response of rails to moving loads is of interest in the area of high-speed railway transportation. If simple geometries of the track and subsoil are considered, a theoretical concept that is based on the assumption that the track structure acts as a continuously supported beam (the rail) resting on a uniform layer of springs can be introduced. This layer of springs represents the underlying remainder of the track structure. The stiffness of such spring layer along the length of the track is named as the track modulus and defines Winkler's model. The Win-kler model is often referred to as a " one-parameter model ". Such a simplified model is traditionally used to estimate the critical velocity of moving trains. The first solution of steady-state dynamic response of an infinite beam on elastic foundation traversed by moving load was presented by Timoshenko [1]. In [2], the moving coordinate system is introduced to convert the governing equation to ordinary differential equation that can be solved by Fourier integral transformation. In [3], the concept of the dynamic stiffness matrix is implemented. Two semi-infinite beams are solved and connected by continuity equations. Then the critical velocity can be determined as the velocity that ensures the nullity of the determinant of the dynamic stiffness matrix. The critical velocity of the load cr v is defined as the phase velocity of the slowest free wave, which in this case is the one that in undamped case induces infinite displacements directed upward as …