Abstract The problem of time dependent variable magnitude travelling loads in a closed symmetric structure is considered. This includes the circumferential propagation in viscoelastically supported cylindrical rings which can rotate about their axes. The main emphasis of the work is to establish the effects of time dependent load variability on the response characteristics when travelling velocities are in the sub-, trans- and supercritical speed ranges. This will enable the Doppler-like interactions among the propagating bending and in-plane waves at the various travelling load speed ranges to be evaluated. As an example simulation, the model is tuned to define the dynamic response characteristics of tires rolling on bumpy surfaces. This enables the sub-, trans- and the supercritical response characteristics to be evaluated.