# Seismic Isolation via I-Shaped and T-Shaped Large-Scale Phononic Metamaterials
## 3 Results  
### I-Shaped LSPMs  
The first structure that was examined was the rectangular trench that forms an I-shaped LSPM. Preliminary numerical calculations showed that the structure becomes more effective when the length of the trench increases, and therefore the varying parameter for the optimization strategy of this LSPM was its width w, while the length was selected as L = 9.0 m. The calculations showed that this structure has wide PBGs for all examined cases at a very low frequency range, although these gaps are found only along the \(\Gamma\)X direction. The numerical results for the examined cases are collected in Table 1, where the calculated lowest (f\({}_{\rm i}\)) and highest (f\({}_{\rm h}\)) frequency limits of each PBG are listed. The gap over mid-gap ratio \(\left(\Delta f/f_{\rm g}\right)\), where \(\Delta f=f_{\rm h}-f_{\rm i}\) and \(f_{\rm g}=(f_{\rm h}+f_{\rm i})/2\), was also calculated and listed in Table 1 as a measure of the width of each calculated gap. For the case where \(\rm w=1.0\) m, two PBGs were calculated. The first between 1.90 and 3.26 Hz with \(\Delta f/f_{g}=0.53\), and the second between 3.97 and 4.41 Hz with \(\Delta f/f_{g}=0.11\). When \(\rm w=2.5\) m, the two calculated PBGs were between 1.74 and 3.21 Hz with \(\Delta f/f_{g}=0.59\) and between 3.32 and 4.15 Hz with \(\Delta f/f_{g}=0.22\). The last examined case for this structure was for width \(\rm w=4.0\) m, and the two calculated PBGs were between 1.68 and 3.10 Hz with \(\Delta f/f_{g}=0.59\) and between 3.26 and 3.74 Hz with \(\Delta f/f_{g}=0.14\). As the width w of the trench increases, the limits of the PBGs are calculated in lower frequencies without significant change in the PBG width. For the case where \(\rm w=4.0\) m, the band structure with the relevant mode shapes of the displacement field for the lowest and highest frequency of the first PBG is shown in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline
**LSPM** & **Varying Parameter** & **PBG Limits** & **PBG Width** \\ \hline \multirow{6}{*}{I-shaped (L = 9 m)} & \multirow{2}{*}{w (m)} & PBG & \multirow{2}{*}{\(\Delta f/f_{\rm g}\)} \\  & & (TX direction) & \\ \cline{1-2}  & & 1.0 & 1.90–3.26 Hz & 0.53 \\ \cline{1-2}  & & 3.97–4.41 Hz & 0.11 \\ \cline{1-2}  & & 2.5 & 1.74–3.21 Hz & 0.59 \\ \cline{1-2}  & & 3.32–4.15 Hz & 0.22 \\ \cline{1-2}  & & 4.0 & 1.68–3.10 Hz & 0.59 \\ \cline{1-2}  & & 3.26–3.74 Hz & 0.14 \\ \hline \multirow{6}{*}{I-shaped with  coated steel core (L = 9 m)} & \multirow{2}{*}{2.5} & 2.93–3.29 Hz & 0.12 \\ \cline{1-2}  & & 3.31–4.24 Hz & 0.25 \\ \cline{1-2} \cline{2-2}  & & 1.56–2.18 Hz & 0.33 \\ \cline{1-2}  & & 2.73–3.28 Hz & 0.18 \\ \cline{1-2}  & & 3.31–4.17 Hz & 0.23 \\ \cline{1-2}  & & 4.44–4.90 Hz & 0.10 \\ \hline \end{tabular}
\end{table} 
Table 1: Collected results for the case of I-shaped LSPMs and the relevant with the coated steel core. The I-shaped cavities, which are filled with a coated steel core, are also shown in Table 1 and provide significant interest regarding the effectiveness of the structure. The rubber coating had thickness \(\mathrm{d}=1\) m. Although the calculated PBGs are found again only along the \(\Gamma\)X direction and are generally not as wide as the empty I-shaped trench, the case where \(\mathrm{w}=4.0\) m showed four PBGs below the frequency of 5 Hz. The first is between 1.56 and 2.18 Hz with \(\Delta f/f_{g}=0.33\), the second is between 2.73 and 3.28 Hz with \(\Delta f/f_{g}=0.18\), and the third is between 3.31 and 4.17 Hz with \(\Delta f/f_{g}=0.23\). There was also a smaller gap between 4.44 and 4.90 Hz with \(\Delta f/f_{g}=0.10\).