Effective medium modelling of real-world multi-modal metamaterial panels achieving broadband vibroacoustic attenuation

Locally resonant metamaterials can achieve unprecedented vibroacoustic performance by subwavelength distributions of small mechanical resonators on a host structure. Substantial broadband vibroacoustic attenuation can be achieved by multi-modal metamaterial panels, which exploit multiple translational and rotational resonator modes to manipulate the overall bending wave propagation. However, the multi-modal metamaterial concept has been studied only for idealized conditions, such as infinite panel extent and uniformly distributed resonators, limiting practical applicability. Efficient methodologies are still needed to study the behaviour of multi-modal metamaterial panels in real-world scenarios. In this work, this challenge is tackled by developing generalized effective medium models, i.e., homogenized material representations through equivalent macro-scale properties, tailored for finite-sized multi-modal metamaterial panels. For the special but important case of simply supported rectangular panels with uniformly distributed resonators, a dedicated analytical effective medium model is developed. For arbitrary boundary conditions and resonator distributions, effective medium finite elements are formulated. The diffuse sound transmission loss (STL) performance is efficiently predicted through Deterministic - Statistical Energy Analysis (Det-SEA), by coupling the effective medium model of the finite-sized metamaterial panel with a diffuse model of the surrounding sound fields. The proposed prediction approaches are validated against detailed FEM modelling, demonstrating that significant computational reductions are achieved while preserving accuracy. Results showcase that multi-modal metamaterial panels maintain broadband vibracoustic attenuation also when subjected to boundary effects and under partial metamaterial treatment.