Response Solutions for the Vibration of a Traveling String on an Elastic Foundation

This Tech Brief presents solutions to the response problem for the vibration of an axially-moving string that is supported by an elastic foundation. This system is of technical interest in the area of flexible media which translates at a high speed, and which is guided by air bearings or similarly modeled distributed supports. The equation of motion is dispersive and contains a skew-symmetric “Coriolis” acceleration component which derives from axial translation of the string. The equation of motion is written in the standard form for a continuous gyroscopic system, so that the string’s stability and response can be analyzed within this broader context. Available modal analysis and Green’s function methods then provide closed form expressions for the response to arbitrary initial conditions and excitation.