# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 1 Introduction
Attracted by its low-frequency band-gap performance, the LR periodic structure has been widely studied by researchers in recent years and various configurations of plate-type structures have been proposed. As summarized by [PERSON] et al. [33], several types of configurations have been proposed to meet the requirements of practical applications, including periodic resonators inserted in the periodic openings of a plate [34; 35; 36; 37; 38], mounted at the surface of a homogeneous plate [39; 40; 41], and also embedded in a close plate with great thickness [42]. Most studies in the available literature about LR structures are related to a single-layer plate, while very few are related to composite laminate structures, owing to their complexity. As the core of a sandwich panel typically has great thickness, it is possible to insert resonators into the core. With the urgent need for vibration control in sandwich panels, [PERSON] and [PERSON] [43] were the first to introduce spring-mass resonators to a sandwich beam in order to study its unusual dynamic behavior with two proposed theoretical continuum models. The initial work was followed and extended by their research group [44; 45; 46; 47; 48; 49] with a specific focus on new theories, practical experiments, the combined effects of periodic resonators and periodic cores, and also the improvement from a single resonator to multiple resonators. These studies were limited to vibrational dynamic behavior and one-dimensional sandwich structures, where acoustic radiation was not considered and cross-stream mode induced by multi-dimension was also not involved. However, in practical applications, most sandwich panels are designed in two dimensions, and sometimes the vibration-induced sound radiation is the major concern except vibration response. Thus, in this paper, the LR sandwich beam is extended to a sandwich plate to meet the practical needs of applications. In addition, the main purpose of this paper is to study the acoustic radiation behavior. For an LR periodic structure, most studies have focused on the effects of the band-gap on vibrational dynamic behavior. However, the effects of the band-gap on sound radiation cannot be represented by or inferred from the effects on vibration, as the sound radiation is not only dependent on vibration response, but also dependent on radiation efficiency. Thus, the study of vibro-acoustic performance of a sandwich plate with periodically inserted resonators has practical meaning. In this paper, the vibro-acoustic performance of an LR sandwich plate is examined, with specific focus on sound radiation performance caused by periodic resonators. A closed-form solution for an LR sandwich plate is theoretically derived. The dispersion curve and the band-gap are obtained by using governing equations for a sandwich plate and Bloch boundary conditions. Then the vibrational response of a finite LR sandwich plate is given to examine the effects of periodic resonators on vibrational performance. In the following, the vibration-induced sound radiation power and corresponding radiation efficiency are studied to examine the acoustic performance of an LR sandwich plate. The variation of acoustic performance is physically explained by the effective mass, and the effect of damping of the resonator to provide more beneficial acoustic performance is also discussed. The findings in this paper provide guidance for band-gap design in an LR sandwich plate when the purpose is not only on the reduction of vibration but also on the reduction of sound radiation.