Moving mass problem: complete solution with the effect of initial conditions

In this contribution, recently published new semi-analytical solution for the moving mass problem [1] is extended to account for the transient terms that adapt the initial part of the complete solution in a way to match the initial conditions. It is assumed that a mass and a vertical force with harmonic component move by constant velocity along a horizontal infinite beam posted on a two-parameter visco-elastic foundation. The new semi-analytical solution is presented as a sum of truly steady-state terms, harmonic terms induced by the moving mass and transient terms adapting the initial conditions. Closed-form formula is given for the first two types of vibrations. It is concluded that transient terms have in most cases almost negligible effect on the full solution and that the initial conditions can significantly affect the amplitudes of the induced harmonic vibrations, but the induced frequencies are kept without any changes.