Coupled Flexural and Torsional Vibration Attenuation With Acoustic Black Hole Metamaterial

Phononic Crystals and elastic mechanical metamaterials have shown some unusual light and mechanical wave manipulating properties. These novel structure-type materials are also finding extensive use in energy harvesting as well as vibration control and attenuation. Phononic Crystals are mechanical metamaterials which are composites of two or more linear elastic materials of different mechanical properties, having vibration energy propagation inhibition over certain range of frequencies. These are called phonic band gaps, whose presence essentially signifies the reduction of flow of phonons in the form of elastic wave energy. Phononic crystals are generally constructed with some periodicity in geometry or boundary so that they are lightweight with a choice of tailoring their mechanical and dynamic properties. Some three dimensional recently developed phononic crystals with Acoustic Black Holes (ABH) also show flexural wave energy localisation inside a structure.
The present study concerns with the finding of wave energy localisation in an elliptic ABH engraved square periodic plate and the generation of locally resonant dispersion curves. Flexural band gaps are observed from the displacement component plots of the structure after simulation. The square periodic plate was next studied with extended external beam surfaces with added masses on the beams. The dispersion curves were plotted and the displacement component plots show torsional wave propagation modes. Now these two structures were combined to form a novel structure comprising of an elliptic ABH engraved square periodic plate with extended beams with masses on them. Coupled bending and torsional band gaps are seen in the structure. The displacement components show attenuation of vibration in torsion as well as flexure.
The numerical analysis was done using Comsol Multiphysics for all the three cases. The elliptic ABH plate is made of Structural Steel, for the torsion and coupled flexure-torsion cases, the plate and beams are made of Aluminium and the masses on the extended beams are made of structural steel. Floquet Theory was used for the Periodic Boundary Condition 1-D and the natural frequencies are found out for each type of unit cell. The frequencies are next swept along the reduced wave vector space to get the dispersion diagram for each unit cell, which show a complete band gap for the flexure case (around 2.5 kHz to 2.7 kHz) and the coupled flexure-torsion case (around 4 kHz to 4.4 kHz). These band gaps are mainly due to the local resonance feature of the unit cells, and was absent in the torsion unit cell. Next, a cantilver beam is constructed with each unit cell. A frequency domain analysis is performed in Comsol and the displacement, velocity and mean-square velocity transmissibility are next plotted in MATLAB for an input point at the mid-point of loaded edge and output point at the mid-point of periodic edge near the fixed edge. Beams are studied with 3, 5, 7 and 10 unit cells in series to ensure the convergence of the frequency response curves. 4 noticeable band gaps are seen for the flexure case around 0.4 kHz to 0.7 kHz, 1.3 kHz to 2.3 kHz, 2.55 kHz to 3.1 kHz and 3.7 kHz to 4.55 kHz. 4 noticeable band gaps are seen for the coupled flexure-torsion case around 0.8 kHz to 1.2 kHz, 2.35 kHz to 3.7 kHz, 3.9 kHz to 4.7 kHz and 5.3 kHz to 7.1 kHz. The present work thus validates the results of band gaps from wave analysis from the Frequency Response band gaps plotted from structural analysis for a novel structure with coupled flexure-torsion vibration modes. Our next aim is to attain band gaps for the torsion structure, for which, study is to be done to modify the geometry or to sweep around higher frequency ranges.