In this paper, we investigate the nonlinear vibrations of a metamaterial structure that consists of a host Euler–Bernoulli beam attached to a periodic array of spring–mass–damper subsystems deployed for vibration absorption. The main objective is to demonstrate that the capability of the metastructure to suppress vibrations can be significantly enhanced when the absorbers are tuned to the optimal frequency. A MATLAB-based optimization algorithm is used to minimize a given objective function for the optimal tuning frequency. A mathematical model is first utilized to perform the linear free and forced vibration analyses. The effect of the local resonators (absorbers) on the suppression of the oscillations of the host beam is studied. The ability to mitigate the vibration of the host structure at a desired resonant frequency is achieved by tuning the resonant frequencies of the local absorbers. More interestingly, the results show that the simultaneous suppression of several modes is possible by tuning and properly placing each absorber along the host structure. Furthermore, the impact of the resonators on the nonlinear behavior of the main structure when subjected to external forcing over an extended frequency range is investigated. The numerical study reveals that proper tuning of the local resonators allows significant vibration suppression of the metamaterial beam when being excited in the neighborhood of the natural frequencies. We demonstrate the capability of the metamaterial structure to withstand external loadings even when operating near resonance. Finally, we combine the nonlinear mathematical model with an optimizer to identify the number and tuning frequencies of the absorbers that maximize the vibration suppression. The optimization results show that significant mitigation can be achieved by tuning properly the absorbers in the vicinity of the host structure’s natural frequencies.