## 3 Examples and Discussion
### Band-Gap Property and Vibration Attenuation
Figure 5: The comparison between the root-mean-square velocities (RMSVs) averaged over the whole surface calculated by the theory and finite element method (FEM) for (**a**) the bare plate, (**b**) the mass-loaded plate, and (**c**) the locally resonant plate. Shown in Figure 6a are the band structure curves of the infinite bare plate, the infinite LR plate, and the infinite ML plate. The RMSVs averaged over the whole plate surface of the bare plate, the LR plate, and the ML plate, for the cases of various boundary conditions, are shown in Figure 6b-d. In the range of 0-500 Hz, the band gaps induced by the LR effect are labeled in Figure 6a, with band-gap frequency ranges of 201-215 Hz and 300-320 Hz, where no waves propagate. The two ranges correspond to the resonant frequencies of two kinds of resonators, i.e., \(f_{R1}~{}=~{}201\) Hz and \(f_{R2}~{}=~{}300\) Hz. It can also be seen that, for the bare plate and the ML plate, waves are allowed at any frequency. The comparison between the band structures in Figure 6a and the vibration results in Figure 6b-d basically show a good coincidence between the band gaps of the infinite LR plate and the vibration attenuation bands of its finite counterpart, no matter what boundary conditions are applied to the finite plate. In the band-gap frequency ranges, the vibration of the finite LR plate is attenuated when compared with those of the bare plate and the ML plate. As illustrated above, the ML plate and the LR plate have the same weight, heavier than the bare plate. From Figure 6b-d, it is obvious that the first few modal frequencies of the bare plate are a little larger than those of the ML plate and the LR plate. Except for the effect on the structural vibration, the weight also affects the acoustic behavior [22]. Thus, only the ML plate is considered for comparison below. It is noted that the RMSV results in Figure 6b-d are all averaged over the whole plate surface, which makes the vibration attenuation in the LR plates not very obvious at some frequencies in two band gaps, especially for the FFFF boundary conditions in Figure 6b. In the finite LR plates, the vibration attenuation in the band-gap frequency ranges is usually evaluated by the vibration transmittance. With the band structure curves of the infinite ML and LR plates shown in Figure 7a for comparison, Figure 7b shows the vibration transmittances of the finite ML and LR plates with FFFF boundary conditions. The vibration transmittance is given by \(T~{}=~{}20\log 10|v_{res}/v_{ext}|\), where the RMSVs averaged over the area of \(0<x<0.1\) m and \(0.5\) m \(<x<0.6\) m within the base plate surface are chosen as \(v_{ext}\) and \(v_{res}\), respectively. From the comparison between the vibration transmittances of the ML and LR plates, it can be seen that the flexural waves in the LR plate can be dramatically suppressed in the band-gap frequency ranges, better revealing its band-gap property. Figure 8a,b shows the displacement profiles of the LR plate at 202 Hz and 310 Hz, respectively within two band gaps. It can be seen that, with the exception of the area around the force position, the excited responses over almost the whole surface of the LR plate are significantly suppressed at the two frequencies. Figure 6: (**a**) Band structures of the infinite bare plate, mass-loaded (ML) plate, and locally resonant (LR) plate, and the root-mean-square velocities (RMSVs) averaged over the whole surface of their finite counterparts with (**b**) fully free (FFFF) boundary conditions, (**c**) fully simply supported boundary conditions (SSSS), and (**d**) fully clamped (CCCC) boundary conditions. (BG: band gap). For all of the above results of the LR plates, the damping characteristic of the resonators is not considered, while it could broaden the vibration attenuation band outside the band-gap frequency ranges [21]. Shown in Figure 9a-c are the RMSVs averaged over the whole surface of the FFFF ML plate and the FFFF LR plate with the \(\eta_{R1}\) and \(\eta_{R2}\) values of the resonators varying separately and simultaneously.