Free and forced vibrations of damped locally-resonant sandwich beams

This paper addresses the dynamics of locally-resonant sandwich beams, where multi-degree-of-freedom viscously-damped resonators are periodically distributed within the core matrix. On adopting an established model in the literature, which consists of an equivalent single-layer Timoshenko beam coupled with mass-spring-dashpot subsystems representing the resonators, novel and exact analytical expressions are presented for the frequency response and modal response under arbitrary loads. Specifically, the frequency response is built by a direct integration method, while the modal impulse and frequency response functions are derived by a complex modal analysis approach, upon introducing orthogonality conditions pertinent to the complex modes. The expressions obtained for frequency response and modal response hold for any number of resonators and degrees of freedom within the resonators, any number of loads and positions of the loads relative to the resonators. In the proposed complex modal analysis approach, the challenging issue of calculating all complex eigenvalues, without missing anyone, is solved applying a recently-introduced contour-integral algorithm to an exact dynamic stiffness matrix that, here, is built with size depending only on the number of degrees of freedom at the beam ends, regardless of the number of resonators and degrees of freedom within the resonators. Numerical applications prove exactness and robustness of the proposed framework.