Steady state response of an infinite string on a non-linear visco-elastic foundation to moving point loads

The steady state response of a long string on a non-linear, visco-elastic foundation to uniformly moving constant point loads is studied. The phase plane is employed to construct a phase portrait that determines the string response. This portrait consists of trajectories of the autonomous system that represent the string displacement everywhere except at the loading points, and vertical segments that satisfy the boundary conditions in these points. It is shown that the string response depends crucially on the ratio between the load velocity and the wave speed in the string. If the load moves slower than the waves in the string, then the response is nearly symmetric with respect to the load and decays exponentially with the distance form the load. If the load velocity exceeds the wave speed, then the string exhibits a wave pattern, which extends well behind the load. In front of the load, in this case, the string is not disturbed.