# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 1 Introduction  
Sandwich structures have been researched for centuries and are one of the most used structures in industry, including but not limited to airplane [1], automobile [2], underwater vehicle [3], and architecture [4]. In general, a sandwich structure usually consists of two high-strength surfaces and one low-strength and also low-density core. Configured by this, sandwich structures can provide an increased stiffness-to-mass ratio compared with both stiffened plates and homogeneous plates when they have the same mass, owing to the increased rotational inertia and bending rigidity for sandwich structures. The stiffness-to-mass ratio plays a very important role in engineering design when the goal of design optimization is reducing weight. As sandwich structures have great potential in application, a quite number of researches can be obtained from the public literatures, with various focus on building theoretical models [5; 6; 7; 8], researching dynamic properties [9; 10; 11], configuring structures with various types of cores [12; 13; 14; 15; 16; 17; 18], and also realizing the engineering applications [1; 2; 3; 4]. In practice, sandwich structures have various configurations for various specific focus, with the main difference lying on the cores. In the aircrafts,honeycomb cores [13; 18] are usually used in order to obtain both enough strength and also light weight. When the core is designed with heat-insulating material [19; 20], the sandwich structures can not only be used in the building walls to maintain temperature of the inner space but also can be applied in a missile to protect the internal precision equipment. Another example is that a sandwich plate with a porous core [15] can be used for noise-isolation purpose. As seen in the above, sandwich structures have many attractive characteristics in the engineering. However, some light-weight sandwich structures may induce acoustic problem, as these light-weight structures vibrate and radiate annoying noise easily. Efforts have been made to improve the poor acoustical performance by using a viscoelastic material core [14], by optimal sandwich design [21; 22; 23], by applying a magnetorheological core or electrorheological core [12; 16], and also by introducing an active surface panel [24; 25]. In recent decades, numerical researches about periodic structures including both phononic crystals [26; 27; 28] and metamaterials [29; 30; 31] have been conducted, as they can effectively attenuate vibration and then reduce noise radiation. Owing to spatial periodicity of periodic structures, waves of some specific frequencies cannot propagate without attenuation while other waves can propagate successfully. In this wave-filtering phenomenon, the frequency band where waves cannot propagate forms a band-gap. In the band-gap, no wave can propagate and thus the vibration can be significantly attenuated. Therefore, periodic structures have steadily come to be considered as a new method to improve vibro-acoustic performance. Researches have shown that Bragg scattering mechanism [32] and locally resonant mechanism [27] are two main mechanisms for the formation of a band-gap. For a locally resonant (LR) periodic structure, the resonance of the resonators is the main cause of the band-gap, and it has been found that the working wavelength of an LR periodic structure can be far greater than the lattice constant (the dimensions of a unit in the periodic structure). Therefore, the LR periodic structure has the ability to reduce lower frequency than a Bragg-scattering structure. [PERSON] et al. [27] first proposed the concept of an LR phononic crystal and verified this type of band-gap with a carefully designed specimen, where lead balls are coated with silicon rubber and they are embedded in an epoxy-matrix plate. By doing this, they obtained an LR band-gap far lower than traditional Bragg band-gap frequencies.