## 3 Results and Discussion
### Shock Impact Testing
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Structure** & \(\rho^{\star}\)/\(\rho_{\text{S}}\)\({}^{\ddagger}\) & **F\({}_{\text{max}}\)\({}^{\ddagger}\)** & \(\uptau\)\({}^{\ddagger}\) & **P\({}^{\ddagger}\)** & **T\({}^{\ddagger}\)** \\  & [-]\(\times\)10\({}^{-2}\) & [N] & [ms] & [N.ms] & [-] \\ \hline CP & 6.60 \(\pm\) 0.05\({}^{\text{a}}\) & 33.6 \(\pm\) 0.3\({}^{\text{a}}\) & 3.08 \(\pm\) 0.03\({}^{\text{e}}\) & 57.9 \(\pm\) 0.9\({}^{\text{a}}\) & 0.136 \(\pm\) 0.001\({}^{\text{a}}\) \\ BCC & 12.10 \(\pm\) 0.24\({}^{\text{b}}\) & 53.8 \(\pm\) 0.9\({}^{\text{b}}\) & 2.95 \(\pm\) 0.03\({}^{\text{e}}\) & 88.3 \(\pm\) 0.7\({}^{\text{c}}\) & 0.211 \(\pm\) 0.002\({}^{\text{b}}\) \\ FCC & 12.12 \(\pm\) 0.79\({}^{\text{b}}\) & 55.7 \(\pm\) 0.9\({}^{\text{b}}\) & 2.66 \(\pm\) 0.01\({}^{\text{d}}\) & 77.8 \(\pm\) 1.0\({}^{\text{b}}\) & 0.216 \(\pm\) 0.001\({}^{\text{b}}\) \\ BCC + CP & 13.6 \(\pm\) 0.58\({}^{\text{c}}\) & 62.8 \(\pm\) 0.8\({}^{\text{c}}\) & 2.41 \(\pm\) 0.01\({}^{\text{c}}\) & 91.2 \(\pm\) 1.0\({}^{\text{c}}\) & 0.238 \(\pm\) 0.001\({}^{\text{c}}\) \\ FCC + CP & 15.8 \(\pm\) 0.71\({}^{\text{d}}\) & 89.5 \(\pm\) 1.0\({}^{\text{d}}\) & 1.55 \(\pm\) 0.01\({}^{\text{b}}\) & 80.1 \(\pm\) 0.8\({}^{\text{b}}\) & 0.341 \(\pm\) 0.002\({}^{\text{d}}\) \\ Solid & 1 & 102.2 e & 1.05\({}^{\text{a}}\) & 53.6\({}^{\text{a}}\) & 0.38\({}^{\text{e}}\) \\ \hline \hline \end{tabular}  
* Remark: \({}^{\dagger}\) There is a significant reference difference at the 95% confidence level. \({}^{\text{a}-\text{e}}\) show the subset order based on the post-hypothesis test.  
\end{table} 
Table 3: Mechanical responses of the different lattice structures. The result from Table 3 (\(F_{\rm max}\)) shows no significant difference between the BCC and FCC at the 95% confidence level. However, the interactions between the relative density and excited force also significantly differed on impulse and maximised at 0.14 the relative density (or BCC + CP). From the force's evolution, it can be seen that the area under the curve of the BCC + CP was the biggest. From this experiment, the possible shock responses in lattice structure were from the FCC + CP \({}^{\rm a}\), BCC + CP \({}^{\rm b}\), BCC \({}^{\rm c}\), FCC \({}^{\rm c}\) and CP \({}^{\rm d}\), respectively. There was no mean difference between the BCC\({}^{\rm c}\) and FCC\({}^{\rm c}\) at the same confidence level denoted by \"C\". Thus, this could lead to the impulse difference from the maximum force graph. Even though the maximum force of these two samples would not have any mean difference between the group, as the data from the FCC were only between 186 \(\pm\) 19 N and 289 \(\pm\) 24 N.  
Even if responses of BCC and BCC + CP have a significant different in maximum transmitting force, it found that the impulse responses were sparingly the same at the homogenous scale \"c\". Thus, impulse responses of these structures may independent on their relative density but it would depend on the architecture of their structures. When an input force increases, the maximum force will nonlinearly increase (see Figure 12). Moreover, the maximised energy absorption structure was the BCC + CP (see Figure 7) with not to fluctuate of stress over the changing of strain that could highlighted the BCC + CP as a good candidate for use in isolation application. The impulse frequency (f, Hz) can be estimated from the inversion of twice the impulse time (1/(2-\(\tau\))). This frequency was a linear function of excited force and mainly had the same trend with the maximum force; hence, all the impulse responses increased linearly with excited force, and depended on the rigidity of the lattice structure, and as the more rigid structures, it could lead to a higher frequency. The impulse frequency of most of the lattices varied roughly between 140-350 Hz (see Figure 13 iii). Although, these frequencies may two time changing, the convolution curve may happen roughly at 5 ms.