The Steady-state Responses And Critical Velocities Of The Infinite Beams On A Viscous Foundation For A Moving Load

Railway train has been a major form of public transportation in the world for more than one and a half centuries. In the last decade, many subways and high-speed railways, including those in Europe, Japan, China, have been constructed or are in construction stage. It's significant to study the response of rails caused by high-speed trains, and the propagation of waves in the rails, for the protection of rails and the security of trains. Beraoulli-Euler and Timoshenko beam models have been used to study the response of the rail under moving loads since the last century, in which only one single moving load has be considered, which is simpler and has no influence to the problem itself. In this paper the steady-state responses and critical velocities of the infinite beams under a moving load will be discussed. All has been done as following:1. Fourier transform is used to solve the problems of steady state response of an infinite beam on a viscous foundation subjected to a moving load. The theorem of residue is employed to evaluate the generalized integral so that an analytic solution can be achieved for the case of constant velocity, to both Bernoulli-Euler beam and Timoshenko beam.2. The influence of damping to the dynamic response of the beam is discussed clearly, which is a key point for solving this problem. The conclusion drawn from the analysis to Bernoulli-Euler beam model can be applied to the solution of Timoshenko beam model directly.3. The characteristics of waves propagated in the Bernoulli-Euler and Timoshenko beam that vibrate freely have been analyzed, as the basis of the solution to the critical velocities.4. The relationship between the critical velocities and resonant frequencies has been obtained, as well the explanations in physics of which.