Linear Dynamics of a Translating String on an Elastic Foundation

This paper examines the free and forced linear response of a string which is translating across an elastic foundation. Exact solutions are derived for the free vibration of the string which translates between fixed eyelets and across elastic foundations represented by (1) a single interior spring and (2) a uniform step foundation. Results illustrate the dependence of the string natural frequencies and mode shapes on the foundation stiffness, the foundation geometry, and the string translation speed. The forced response of the string to harmonic end excitation is computed in closed form for the case of a complete uniform foundation. A cutoff frequency separates three distinct solution forms. For excitation frequencies below the cutoff frequency, the response amplitude decays exponentially with distance from the driven end.