Physical realization and experimental validation of effective phononic crystals for control of radial torsional waves

Recently, the concept of effective phononic crystals (EPCs) and locally resonant effective pho-nonic crystals (LREPCs) have been proposed to enable properties of phononic crystals (PCs) and acoustic metamaterials (AMs), respectively, in radially propagating waves. This problem is not straightforward since the equations of motion of radially periodic systems do not have periodic coefficients and thus Bloch solutions in these radially periodic systems are not valid. To overcome this, EPCs and LREPCs use radially dependent properties to force periodic coefficients in the equations of motion of radial elastic waves, enabling the application of the Bloch theorem and thus allowing for properties of PCs and AMs. However, the required radially dependent properties severely complicate their physical realization, since modulus and density must be tailored such that they follow a specific radial variation. In this paper, we propose a method to physically realize an LREPC by radially varying its impedance through spatial changes to its out-of-plane thickness. Physical realization of the LREPC also involves local torsional resonances. In contrast to typical AMs, the resonators in the LREPC must be geometrically different to retain the same torsional stiffness and moment of inertia at different radii. We use additive manufacturing plus traditional machining to fabricate the LREPC. We introduce an experimental setup to mea-sure radially propagating torsional waves in the LRPEC and show how to decouple torsional vi-brations from bending vibrations to accurately measure transmission. To show the importance of using an LREPC, we compare its dynamic response to a locally resonant homogenous system (LRHS), a system that is radially periodic but has non-periodic equations of motion. Measured transmission shows that only the LREPC attenuates waves inside the band gap frequencies pre-dicted by Bloch analysis. This work experimentally shows that LREPCs allow for the application of well-known concepts of AMs to control radially propagating torsional waves.