Metrikine A.V.

SteadyState Response of a Long Cable on NonLinear ViscoElastic Foundation to a Set of Moving Loads 
Journal of Sound and Vibration
272 (35), 10331046 (2004). 
Summary. The steadystate response of a long string on nonlinear, viscoelastic foundation to a set of uniformly moving constant loads is studied. The phase plane is employed to construct a phase portrait that determines the string response. This portrait consist of trajectories of the autonomous system that represent the string displacement everywhere except 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 almost 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, the string is not disturbed. 