On the free vibrations of locally-resonant structures

This paper addresses the free-vibration response of structures coupled with periodically-distributed resonators, generally referred to as locally-resonant structures. The first step is to show that all exact natural frequencies can be calculated by a proper formulation of the Wittrick-Williams algorithm, involving a condensed dynamic stiffness matrix whose size depends only on the number of degrees of freedom of the structure and is independent of the number of degrees of freedom within every resonator. Indeed, the presence of resonators is accounted for in the condensed dynamic stiffness matrix via a pertinent frequency-dependent stiffness, readily obtainable from the resonator motion equations. Within this framework, a novel procedure is proposed to construct the condensed dynamic stiffness matrix of locally-resonant plates, applicable for any number of resonators. Numerical applications show the exactness of the proposed formulation.