The low-frequency vibration and radiation performance of a locally resonant (LR) plate with periodic multiple resonators is studied in this paper, with both infinite and finite structure properties examined. For the finite cases, taking the LR plate attached with two periodic arrays of resonators as an example, the forced vibration response and the radiation efficiency are theoretically derived by adopting a general model with elastic boundary conditions. Through a comparison with the band structures calculated by the plane-wave-expansion method, it shows that the band gaps in the infinite LR plate are in good agreement with the vibration-attenuation bands in the finite LR plate, no matter what boundary conditions are applied to the latter. In contrast to the vibration reduction in the band gaps, the radiation efficiency of the finite LR plate is sharply increased in the band-gap frequency ranges. Furthermore, the acoustic power radiated from the finite LR plate can be seriously affected by its boundary conditions. For the LR plate with greater constraints, the acoustic power is reduced in the band-gap frequency ranges, while that from the one with fully free boundary conditions is increased. When further considering the damping loss factors of the resonators, the attenuation performance can be improved for both the vibration and radiation of the LR plate.

The vibro-acoustic performance of a sandwich plate with periodic locally resonant (LR) units is examined in this paper, with specific focus on the effect of periodic resonators on the average radiation efficiency and the acoustic radiation to the far field. In order to assess the radiation performance, the band-gap properties of an infinite periodic structure and the vibrational response of a finite periodic structure are first studied with closed-form solutions. Subsequently, the acoustic radiation efficiency of the LR sandwich plate is obtained using the concepts of modal radiation. It is shown that the acoustic radiation power can be reduced significantly, not only in the band-gap but also at frequencies close below the band-gap, due to either the decrease in radiation efficiency or the decrease in the vibration response. Thus, the periodic resonators provide a broader attenuation band for the purposes of noise reduction than for vibration reduction. However, for frequencies close above the band-gap, the acoustic performance became worse, owing to the increase in acoustic radiation efficiency. Fortunately, the increased sound radiation above the band-gap can be reduced by adding a small damping to the resonator, which further broadens the attenuation frequency band. The reason for the variation of acoustic radiation efficiency is also studied and can be physically explained by the effective mass of an LR unit, where increased mass corresponds to decreased radiation efficiency and decreased mass corresponds to increased radiation efficiency. Thus, the effective mass can be a useful parameter for designers to estimate which frequency component will be acoustically reduced or acoustically enhanced in a practical design.