A power flow method for evaluating vibration from underground railways

One of the major sources of ground-borne vibration is the running of trains in underground railway tunnels. Vibration is generated at the wheel–rail interface, from where it propagates through the tunnel and surrounding soil into nearby buildings. An understanding of the dynamic interfaces between track, tunnel and soil is essential before engineering solutions to the vibration problem can be found. A new method has been developed to evaluate the effectiveness of vibration countermeasures. The method is based on calculating the mean power flow from the tunnel, paying attention to that part of the power which radiates upwards to places where buildings’ foundations are expected to be found. The mean power is calculated for an infinite train moving through the tunnel with a constant velocity. An elegant mathematical expression for the mean power flow is derived, which can be used with any underground-tunnel model. To evaluate the effect of vibration countermeasures and track properties on power flow, a comprehensive three-dimensional analytical model is used. It consists of Euler–Bernoulli beams to account for the rails and the track slab. These are coupled in the wavenumber–frequency domain to a thin shell representing the tunnel embedded within an infinite continuum, with a cylindrical cavity representing the surrounding soil.