Flat bands in magic-angle vibrating plates

Twisted bilayer graphene develop quasi-flat bands at specific ``magic'' interlayer rotation angles through an unconventional mechanism connected to carrier chirality. Quasi-flat bands are responsible for a wealth of exotic, correlated-electron phases in the system. In this work we propose a mechanical analogue of twisted bilayer graphene made of two vibrating plates, patterned with a honeycomb mesh of masses, and coupled across a continuum elastic medium. We show that flexural waves in the device exhibit vanishing group velocity and quasi-flat bands at magic angles, in close correspondence with electrons in graphene models. The strong similarities of spectral structure and spatial eigenmodes in the two systems demonstrate the chiral nature of the mechanical flat bands. We derive analytical expressions that quantitatively connect the mechanical and electronic models, which allow us to predict the parameters required for an experimental realization of our proposal.