## 3 Examples and Discussion
### Band-Gap Property and Vibration Attenuation  
It should be noted that introducing a new type of resonator will affect the original band-gap frequency range in the LR plate with one resonator in each periodic element (ORIEPE LR plate), which is related to the spring-mass parameters of the new resonators. Thus, the effect of the ratio between the resonant frequencies on the band-gap frequency range is studied first of all. Since the complete band gaps are closely related to the LR mechanism in the LR plate system, directional band gaps are not considered in this paper. Shown in Figure 3 are the trends of two band-gap frequency ranges versus \(f_{R2}/f_{R1}\) in the TRIEPE LR plate with resonators \((m_{R1},\,k_{R1})\) and \((m_{R2},\,k_{R2})\), with the resonant frequencies of \(f_{R1}\) and \(f_{R2}\), respectively. The band-gap frequency ranges in the two corresponding ORIEPE LR plates are also shown in Figure 3 for comparison. The parameters \(m_{R1}\ =\ 0.020\,\text{kg}\), \(k_{R1}\ =\ 3.2\times 10^{4}\,\text{N}/\text{m}\), \(f_{R1}\ =\ 201\text{Hz}\), and \(m_{R2}\ =\ 0.014\,\text{kg}\) are kept unchanged, while \(f_{R2}/f_{R1}\) is varied by changing \(k_{R2}\). It can be seen in Figure 3 that the single band gap in the ORIEPE LR plate with \((m_{\rm R1}\), \(k_{\rm R1})\) is 201-233 Hz, as shown in the dark grey shaded area. The light grey shaded area shows that, within the frequency range of 0-500 Hz, with the increase of \(f_{\rm R2}/f_{\rm R1}\), both the lower and upper frequencies of the single band gap in the ORIEPE LR plate with \((m_{\rm R2}\), \(k_{\rm R2})\) keep increasing, and the lower frequency is very close to the resonant frequency \(f_{\rm R2}\). When \(f_{\rm R2}/f_{\rm R1}~{}=~{}1\), the lower frequencies of the two single band gaps are totally the same, while the difference between \(m_{\rm R1}\) and \(m_{\rm R2}\) makes the upper frequencies 233 Hz and 224 Hz, respectively, which means the resonator mass plays an important part in the determination of the upper frequency. Furthermore, it can be seen that the lower frequencies of the two single band gaps in both ORIEPE LR plates almost coincide with those of the two band gaps in the TRIEPE LR plate, while the band gaps in the former cases cover those in the latter case. In the TRIEPE LR plate with both \((m_{\rm R1}\), \(k_{\rm R1})\) and \((m_{\rm R2}\), \(k_{\rm R2})\), when \(f_{\rm R2}/f_{\rm R1}<1\), the band gap formed by the resonance of \((m_{\rm R2}\), \(k_{\rm R2})\) is the 1 st band gap. The 1 st band gap is getting narrower as \(f_{\rm R2}\) approaches \(f_{\rm R1}\), and meanwhile the 2 nd band gap is getting wider. When \(f_{\rm R2}/f_{\rm R1}~{}=~{}1\), the original 1 st band gap vanishes and only one band gap exists. As \(f_{\rm R2}/f_{\rm R1}\) continues increasing, two band gaps reappear, while the band gap formed by the resonance of \((m_{\rm R2}\), \(k_{\rm R2})\) turns into the 2 nd band gap. The bandwidth of the 1 st band gap is not changed after a slight increase, and the bandwidth of the 2 nd band gap keeps increasing after a decrease. Shown in Figure 4 are the total bandwidths of the band gaps in three corresponding LR plates when \(f_{\rm R2}\) is below 500 Hz. The bandwidth of the band gap in the ORIEPE LR plate with \((m_{\rm R1}\), \(k_{\rm R1})\) is 32 Hz. With the increase of \(f_{\rm R2}/f_{\rm R1}\), the bandwidth in the ORIEPE LR plate with \((m_{\rm R2}\), \(k_{\rm R2})\) has a nearly linear increase, which makes its bandwidth curve intersect with that in the ORIEPE LR plate with \((m_{\rm R1}\), \(k_{\rm R1})\) when \(f_{\rm R2}/f_{\rm R1}\) reaches a certain value. Moreover, the curve of the total bandwidth of the two band gaps in the TRIEPE LR plate seems positively proportional to \(f_{\rm R2}/f_{\rm R1}\) with a lower slope than that in the ORIEPE LR plate with \((m_{\rm R2}\), \(k_{\rm R2})\). Interestingly, three bandwidth curves intersect at \(f_{\rm R2}/f_{\rm R1}\approx 1.4\), where the total bandwidths are all 32 Hz.