Wannier-Stark ladders in free oscillations of Timoshenko-Ehrenfest beams

Abstract

Abstract We report on the observation of Wannier-Stark ladders in the bending vibrations of elastic beams. By introducing a gradient in the length distribution of N weakly coupled beams, the elastic equivalent of the Wannier-Stark ladder is obtained in a system governed by two coupled second-degree differential equations, instead of the common wave equation, and whose oscillations are also dispersive and, above a certain critical frequency, occur with two wavelengths. We have measured for the first time, not only the spectrum of the ladders, but the wave amplitudes in this type of system, which are not directly accessible in quantum-mechanical systems, and we have found that the wavelengths are spatially localized and in good agreement with the theoretically predicted amplitudes. Due to the combination of differential equations and the boundary conditions imposed on the system, the Wannier-Stark ladder phenomenon occurs from the third band of states onwards, unlike other classical systems.