# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 2 Formulations  
### Structure Model and Governing Equations  
A unit element of a periodic LR sandwich plate is shown in Figure 1, where a resonator consisting of a mass block and two linear springs is mounted in a cylindrical hollow in the core layer of a symmetric sandwich plate. In this coupled system, the mass block is connected to the top and bottom surface layers with two identical linear springs. Thus, in the LR band-gap, the vibration energy of the surface panels can be transmitted to the inner mass and most of the system energy will be stored in the periodic resonators. The change in vibration distribution and the change in radiation efficiency induced by coupling between the sandwich panel and its inserted periodic resonators will lead to a change in the corresponding acoustic radiation. The unit element of the LR sandwich plate has the dimensions of length \(a_{1}\), width \(b_{1}\), surface-layer thickness \(h_{f}\), and core thickness \(h_{c}\). The material parameters of the surface layer are the Young's modulus \(E_{f}\), density \(\rho_{f}\), and Poisson's ratio \(v_{f}\), while the material parameters of the core layer are the shear modulus \(G_{c}\) and density \(\rho_{c0}\). Each linear spring has identical stiffness \(k_{R}/2\) and the mass of the resonator is denoted as \(m_{R}\). In addition, the radius of the inner cylindrical hollow is \(r_{o}\). In the following theoretical model, an inhomogeneous sandwich plate with periodic cylindrical hollows is modeled as a homogeneous sandwich plate with an equivalent density of the core expressed as \(\rho_{c}=\rho_{c0}\big{(}1-\pi r_{o}^{2}/a_{1}b_{1}\big{)}\) and the resonator is assumed to be attached under the homogeneous sandwich plate with a mass \(m_{R}\) and a spring stiffness \(k_{R}\). In addition, the shear moduli of the surface layers are dramatically larger than that of the inner core. Thus, the surface layers are assumed to  
Figure 1: (**a**) Configuration of a unit element and (**b**) the first Brillouin zone. suffer longitudinal strain but no transverse shear strain, while the core layer suffers shear strain but no normal strain. In addition, the surface layers and the core layer are configured in perfect continuity and at the interfaces between each layer no slip occurs. The transverse displacements in each layer keep identical at the same in-plane position. Harmonic motion with a time-dependence term \(e^{j\omega t}\) is considered throughout the paper. Following the above assumptions, the strain energy of the surface layer and the core layer can be expressed respectively as  
\[ \begin{array}{ll}U_{f}&=\frac{1}{2}B\int\int\left[u_{x}^{2}+2v_{f}u_{x}v_{y} +v_{y}^{2}+v_{a}\big{(}u_{y}+v_{x}\big{)}^{2}\right]dxdy\\ &+\frac{1}{2}D\int\int\left[w_{xx}^{2}+2v_{f}w_{xx}w_{yy}+w_{yy}^{2}+2\big{(}1- v_{f}\big{)}w_{xy}^{2}\right]dxdy\end{array} , \tag{1}\]  
and  
\[U_{c}=\frac{1}{2}G_{h}\int\int\left(\gamma_{x}^{2}+\gamma_{y}^{2}\right), \tag{2}\]  
where \(u\), \(v\), and \(w\) are the \(x\)-wise, \(y\)-wise, and \(z\)-wise surface displacements, respectively. The terms \(\gamma_{x}\) and \(\gamma_{y}\) are the shear strains of the sandwich core and are expressed, respectively, as \(\gamma_{x}=(2u-dw_{x})/h_{c}\) and \(\gamma_{y}=\big{(}2v-dw_{y}\big{)}/h_{c}\), where \(d=h_{c}+h_{f}\). In addition, \(B=E_{f}h_{f}/1-v_{f}^{2}\), \(D=E_{f}h_{f}^{3}/12\Big{(}1-v_{f}^{2}\Big{)}\)\(v_{a}=\big{(}1-v_{f}\big{)}/2\), and \(G_{h}=G_{c}/h_{c}\).