Natural flexural vibrations of a continuous beam on discrete elastic supports

It has been suggested that modern rail systems might exploit so-called floating tracks, to minimise traffic vibration and noise. This paper discusses the transverse deflexion of an infinite Bernoulli–Euler beam mounted on discrete elastic supports, a model considered suitable to explore low-frequency vibrations and associated resonances in such systems. The dynamics is governed by the eigenvalues and eigenvectors of a transfer matrix, which relates the deflexion of any beam span to the deflexions of its neighbours. Important “extensive” contributions, rather than “spatially damped” modes, occur whenever the transfer matrix has one or more eigenvalues of modulus 1. Responses such as the so-called “pinned–pinned resonance” occur when these eigenvalues of modulus 1 are real (i.e., the eigenvalues are ∓1); and further modes corresponding to two complex conjugate eigenvalues coalescing into ∓1 arise at other wavelengths, when the supports are elastic—i.e., in addition to the resonant modes identified in many earlier analyses assuming fixed supports. There is no average energy flux from span to span for any mode defined by a real eigenvector, and we infer that zero-energy transfer between spans is a characteristic of the resonant response of the system to a stationary vibrating source located on some particular span.