Bending waves in quasiperiodic beams and plates

Abstract

Flexural waves in slender structures, beams or plates, take an important place in engineering applications since they are easily excited and play a dominant role for noise generation. The reduction of flexural motion is an old problem, be it for building materials, automotive applications, or precision instruments. The vast majority of metamaterial studies in the vibroacoustic domain is aiming at vibration reduction of one- and two-dimensional structures. Their models are very often based on the periodicity assumption, giving physical insights with minimal calculation power. However, nature shows us that materials without translational symmetry in their structure can give rise to exotic wave scattering behaviour, very similar to Bragg scattering but with different rotational symmetries leading to 5- and 10-fold diffraction patterns. At the same time, Penrose showed mathematically that planes can be filled with combinations of a limited number of polyhedra in a fully aperiodic way. Despite the absence of any translational periodicity, these structures are uniquely defined and have recognizable local rotational symmetries. They are therefore called quasicrystals. We present two examples of bending waves in quasicrystalline structures, both numerical and experimental. The first case shows unexpected low-frequency band gaps in beams with two periodic arrays of slits. The scattering of bending waves due to the aperiodic changes in bending stiffness results in interaction between the two periodic Bragg band gaps. The second example shows the formation of band gaps and localized modes in a quasicrystal plate dressed with scatterers in a Penrose pattern. Band gaps occur at lower frequencies than in a periodic array of scatterers with the same density, but is therefore less efficient. In both cases the structures showcase areas with low and high vibrational amplitudes, a fact that might be exploited to lead high energy concentrations away from critical points.