Optimal resonator damping for wave propagation control in mechanical metamaterials

A novel approach to the optimal damping of linearly damped resonators embedded in metamaterial systems is proposed with the aim of minimizing the metamaterials' response when the external excitation frequency lies within one of the bandgaps. The equation governing wave propagation in the metamaterial system is obtained via a Galerkin projection combined with the quasi-periodicity ansatz of the Floquet-Bloch theorem. It is shown that an optimality criterion can be obtained for the resonator damping for any excitation frequency by extending the Den Hartog theory of fixed points in the frequency response functions. A numerical example of a honeycomb metamaterial is discussed to show how the proposed method works in a practical application. A full numerical optimization is carried out to study the quality factor of the metamaterials' response with respect to the resonator damping ratio while proving its effectiveness.