On the Free Vibrations of Non-Classically Damped Locally Resonant Metamaterial Plates

In this paper, the focus is on the free vibrations of locally resonant metamaterial plates with viscously damped resonators. Upon formulating a dynamic-stiffness model where the resonators are represented via pertinent reaction forces depending on the deflections of the attachment points, the complex eigenvalues are calculated by a contour-integral algorithm introduced in the literature for general nonlinear eigenvalue problems. The interest in the proposed approach is twofold. The dynamic-stiffness model involves a limited number of generalised coordinates compared to the nodal degrees of freedom of a standard finite-element model, and the contour-integral algorithm proves successful in evaluating all complex eigenvalues, without missing any one, with remarkable computational efficiency. Numerical results are presented for Lévy plates, but are readily extendible to other plate theories. Finally, an ad hoc dynamic-stiffness approach is formulated to calculate the frequency response of the plate under arbitrarily placed loads, which is of particular interest to investigate its elastic wave attenuation properties.