# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 3 Results and Discussion
### Vibration Response of a Finite LR Sandwich Plate  
The vibration response of a finite LR sandwich plate with a simply-supported boundary is further studied to illustrate the band-gap phenomenon of vibration in a practical finite structure. The examined structure is composed of \(10\times 8\) cells (\(1\times 0.8\) m) with cell parameters the same as those shown in the previous sub-section. A transverse point force is applied at position (0.1, 0.1 m) with a harmonic form and the whole structure is structurally damped with a damping loss factor of \(\eta=0.002\).  
The root-mean-square velocity averaged over the plate surface and each modal contribution of an LR sandwich plate are shown in Figure 5a, together with the results of an equivalent homogeneous sandwich plate shown in Figure 5b. Comparison of Figures 5a and 3 shows that the band-gaps of an infinite periodic LR sandwich plate are coincident with the response valleys of a finite periodic structure. In the band-gap frequency region (868-1032 Hz), the vibration of the LR sandwich plate is significantly reduced compared with that of a homogeneous sandwich plate.  
In fact, the reduction of vibration in the band-gap can be explained by the effects of the resonators on each vibration mode. As can be observed in Figure 5a, all the modal responses are significantly affected by periodic resonators, with a weak response contribution in the band-gap but strong contribution outside of the band-gap. Further explanation is given in Figure 6, where three specific modes (mode (2, 3), mode (5, 4), and mode (5, 6)) are given for both the LR and homogeneous sandwich plates. It is observed that, for the homogeneous sandwich plate, only one peak appears in the curve, which corresponds to the flexural vibration mode. However, when the resonators are introduced to the sandwich plate, a new peak related to the resonator vibration mode is generated as a result of vibration coupling between the resonator and base plate. In addition, the flexural mode of the homogeneous sandwich plate is shifted to a lower or higher frequency in the LR sandwich plate.  
Figure 5: Root-mean-square velocity (RMSV) averaged over the plate surface and corresponding modal contributions in (**a**) a locally resonant sandwich plate and (**b**) a homogenous sandwich plate. (BG—band-gap.)  
Further observation of Figures 5 and 6 shows that these two modal frequencies in the LR sandwich plate are always set apart by the band-gap, with one mode positioned at the left side and the other at the right side, and that no mode exists in the band-gap. This phenomenon is also revealed through modal analysis of an LR sandwich plate and a homogeneous sandwich plate, and their modal frequencies are shown in Figure 7. For the homogeneous sandwich plate, the modes may be positioned at any frequency based on the dimensions and boundary conditions of the finite structure. However, for a homogenous sandwich plate, when resonators are introduced the modal frequencies positioned within the band-gap are shifted to the left side or right side of the band-gap. As a result, vibration modes are not allowed in the band-gap of the LR sandwich plate. Thus, the vibration in the band-gap can be reduced.