# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Band-Gap Property  
The dispersion curves of an infinite LR sandwich plate are calculated using the theory derived in Section 2.2.1. At the same time, the dispersion curves of an infinite homogeneous sandwich plate with the same dimensions and same materials are also discussed for comparison purposes. The dispersion curves of these two structures are shown together in Figure 3. It can be seen that, for the homogeneous sandwich plate, each frequency is always related to a wave number in the \(\Gamma\)X direction and a wave number in the \(\Gamma\)X direction in the irreducible Brillouin zone, which means that propagating modes are allowed at any frequency. However, for the LR sandwich plate, there are some frequencies that are not related to any wave number in the \(\Gamma\)X direction. Two directional band-gaps (DBG) along the \(\Gamma\)X direction and one complete band-gap (CBG) are observed in the figure with band-gap frequencies of 825-1032 Hz and 1476-1677 Hz for the two DBGs and 868-1032 Hz for the CBG, respectively. Further analysis shows that the first DBG located near the resonance of the resonator acts as a resonance gap, which is caused by the LR effect; and the second DBG is a Bragg gap, which is caused by wave scattering at periodic discontinuities and is affected significantly by the lattice constant. These two types of band-gaps can be identified by comparing the modal shape of each dispersion curve. The Bragg gap can also be identified by the Bragg frequency \(f_{B}\) (the lower frequency edge of a Bragg gap), where the wavelength is twice as that of the lattice constant, and it can be determined by solving the transcendental equation \(k_{P}(f_{B})=\pi/a_{1}\), where \(f_{B}\) is calculated to be 1476 Hz. In a finite periodic structure, the DBGs along the \(\Gamma\)X direction can only exist in a situation with symmetric excitation, whereas the CBG is more general, where a wave from any direction can be attenuated and the sound radiation from a finite structure is affected significantly by the CBG instead of the DBG. Thus, this research will focus on the CBG and the term \"band-gap\" used in the next sections will refer to the CBG. The theoretical formulations are validated using a finite-element-method (FEM) numerical model based on the COMSOL Multiphysics software. Shown in Figure 4a is a unit element, which mainly consists of the bottom sandwich part and the top mass-block part. The dimensions and material properties of the sandwich part are the same as that shown in the beginning of Section 3, except that the density of the core is modified to be 102.2 kg/m\({}^{3}\) as the core layer is considered as homogeneous in the FEM model. The mass of the top mass-block part is set to be 0.0315 kg and its Young's modulus is set to be as large as \(1\times 10^{15}\) Pa so that the block could be considered as a rigid body. A connection function called \"elastic layer\" in COMSOL is added in the FEM modal between the bottom sandwich part and the top mass-block part, so that the two parts are elastically connected. The spring stiffness of the \"elastic layer\" is set to be \(k_{R}=1.2436\times 10^{6}\,\mathrm{N/m}\), which is the same as that described in the beginning of Section 3. The meshing model is shown in Figure 4b with the element size set as small as 2 mm for the purpose of satisfying the adequate meshing resolution at the highest frequency 2000 Hz. In order to obtain the band-gap properties, the Bloch-Floquet periodic conditions are applied at two in-plane (both x-wise and y-wise) directions in the FEM model, which are represented by a Bloch wave vector \(\mathbf{k}=(\alpha,\beta)\). By sweeping \(\mathbf{k}\) in x-wise direction (\(\alpha:0\rightarrow\pi/a_{1}\,,\beta:0\)), y-wise direction (\(\alpha:\pi/a_{1}\,,\,\beta:0\rightarrow\pi/a_{2}\)), and corner-to-corner direction (\(\alpha:\pi/a_{1}\to 0\,,\,\beta:\pi/a_{2}\to 0\)), the dispersion relationship between wave number and frequency can be obtained by further applying characteristic frequency analysis in the simulation software.