Contrast parameter characterizes Bragg frequency gap in periodic flexural beams

Abstract

Periodic structures exhibit frequency bands where destructive interferences prohibit wave propagation. Such behavior can be useful to mitigate vibrations, in particular when structure lightening is important. Band gap properties can be deduced from Bloch's theory approach, Plane Wave Expansion or Multiple Scattering methods. These methods require a full description of the unit cell geometry and its mechanical properties. In this work we focus on an ideal unit cell geometry to highlight the role of a contrast parameter in the band gap opening process. We consider flexural waves in beams and demonstrate analytically that the contrast parameter fully controls the first Bragg band gap. Numerical simulation and experiments on a beam demonstrator proves that the gap bandwidth is independent of the section geometry: only the flexural rigidity is involved. We propose a semi-analytic model for the central frequency gap that depends on the mass distribution. The established algebraic expressions for the band gap bandwidth and central frequency successfully matches the results in the practical case and can be used to design flexural wave cut-band filters. Finally, symmetry considerations explains the experimental observation of a second band gap, also suitable for vibration mitigation, due to coupling of flexural and compressional waves.