## 4 Conclusions  
In this paper, the low-frequency vibration and radiation performance of an LR plate is analyzed. The theoretical model is established for the band structures of the infinite LR plate with periodic multiple resonators. Taking the finite LR plate attached with two periodic arrays of resonators as an example, the forced vibration response and the radiation efficiency of the finite LR plate are derived, by adopting a general model with elastic boundary conditions. First, we discuss the impact of the parameter of the ratio between the resonant frequencies of two resonators on the band-gap frequency ranges. The band-gap width in the TRIEPE LR plate stays between those in both corresponding ORIEPE LR plates. By using the proper spring-mass parameters, the band structures of the infinite bare, ML, and LR plates and the RMSVs of their finite FFFF, SSSS, and CCCC counterparts are calculated. The comparison shows a good coincidence between the band  
Figure 14: The displacement profile of the locally resonant plates with damped resonators with fully free boundary conditions at 304 Hz. Figure 13: The displacement profiles of (**a**) the mass-loaded plate and (**b**) the locally resonant plate with fully free boundary conditions at 304 Hz. gaps in the infinite LR plate and the vibration attenuation bands in the finite LR plate, no matter what boundary conditions are applied to the latter. A better vibration attenuation performance can be achieved when further considering the damping characteristic of the resonators. In contrast to the obvious decrease of the RMSV, the radiation efficiency of the finite LR plate has been sharply increased in the band-gap frequency ranges compared with the ML plate. By comparing the structural and acoustic wavenumbers, it can be seen that the acoustic radiation from the infinite LR plate is effective in the band gaps, and correspondingly that its finite counterpart can radiate sound highly effectively in the band-gap frequency ranges. Moreover, the acoustic power radiated from the finite LR plate around the band-gap frequencies may be seriously affected by its boundary conditions. The acoustic power radiated from the FFFF LR plate could be increased in the band-gap frequency ranges due to no effective cancellation of radiation from the strongly vibrating area near the excitation. However, for the LR plate with greater constraints, such as the SSSS and CCCC boundary conditions considered in this paper, the acoustic power is decreased in the band-gap frequency ranges. When further considering the damping characteristic of the resonators, a better vibration attenuation performance can result in the improvement of a radiation reduction around and above the band-gap frequencies. Conceptualization, Q.Q. ; Data curation, M.S. ; Formal analysis, Q.Q. ; Investigation, M.S. and Z.G. ; Methodology, Q.Q. ; Resources, M.S. ; Software, M.S. and Z.G. ; Supervision, M.S. ; Validation, Q.Q., M.S. and Z.G. ; Visualization, Z.G. ; Writing--original draft, Q.Q. ; Writing--review & editing, M.S. and Z.G. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. The terms \(\mathbf{B}_{a0}\), \(\mathbf{B}_{a1}\), \(\mathbf{B}_{a2}\), \(\mathbf{B}_{a3}\), \(\mathbf{B}_{a4}\), \(\mathbf{B}_{b0}\), \(\mathbf{B}_{b1}\), \(\mathbf{B}_{b2}\), \(\mathbf{B}_{b3}\), and \(\mathbf{B}_{b4}\) in Equation (7) can be expressed as  
\[\mathbf{B}_{s0}\ =\ \left[ \begin{array}{cccc}0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0\end{array} \right], \tag{10}\]  
\[\left\{ \begin{array}{ll}\mathbf{B}_{s1}&=\ diag(1,-1,3,-3)\cdot\pi/2/s,\\ \mathbf{B}_{s2}&=\ diag(-1,-1,-9,-9)\cdot\pi^{2}/4/s^{2},\\ \mathbf{B}_{s3}&=\ diag(-1,1,-27,27)\cdot\pi^{3}/8/s^{3},\\ \mathbf{B}_{s4}&=\ diag(1,1,81,81)\cdot\pi^{4}/16/s^{4}\end{array} \right., \tag{11}\]  
where \(s\) in the above equations represents correspondingly \(a\) or \(b\).