A reduced-order framework for airborne and impact sound transmission analysis of finite metamaterial structures

Resonant metamaterials are a promising strategy to enhance vibroacoustic attenuation in lightweight structures. Their typically periodic layout allows compact representations by discretizing one single unit cell (UC) with finite elements (FE). Since resolving the detailed geometry and subwavelength dynamics of the UCs often requires large FE models, UC model order reduction techniques, such as Generalized Bloch Mode Synthesis, have been introduced. Boundary condition effects are then captured by assembling the reduced UC models into a global reduced model. However, the analysis of the interaction between these finite-sized metamaterial structural models and surrounding diffuse sound fields is challenging. In this work, a solution is pursued by embedding the global reduced metamaterial structural model within the deterministic–statistical energy analysis framework. Here, the deterministic description of the metamaterial structure is coupled with a statistical subsystem formulation of the diffuse acoustic fields by exploiting the reciprocity relationship between direct field radiation and diffuse reverberant loading. The methodology is applied to evaluate the airborne sound transmission and impact sound radiation performance of a metamaterial panel consisting of an orthotropic cross-laminated timber plate (CLT) with integrated multi-modal rotational resonators. These generate two directional bandgaps and suppress coincidence effects along the principal axes of the CLT plate. The proposed modeling approach shows excellent agreement with laboratory sound transmission and radiation measurements, confirming its accuracy and efficiency in the analysis of lightweight metamaterial structures.