# Combined Attenuation Zones of Combined Layered Periodic Foundations
## 3 Results and Discussion
### Combined Attenuation Zones of CLPFs  
Figure 6 shows FRF of the CLPF with \(N_{1}\) = 2 and \(N_{2}\) = 2, where the geometric parameters of I-type unit cell are \(a\) = 0.5 m and \(f_{r}\) = 30% while those of II-type unit cell are \(b\) = 1 m and \(f_{r}\) = 30%. For comparison purposes, the corresponding AZs of the traditional LPFs with I-type and II-type unit cells obtained by Equation (6) are plotted as the blue shaded areas (the first two AZs are 4.67-15.50 Hz and 16.88-30.99 Hz) and grey shaded areas (the first two AZs are 2.33-7.75 Hz and 8.44-15.50 Hz) in Figure 6, respectively. The green shaded areas are the so-called CAZs, which are a union of AZs of the traditional LPFs with I-type and II-type unit cells. It can be seen from Figure 6 that the frequency ranges of negative FRFs match well with the CAZs except for frequencies 2.63, 4.07 and 7.86 Hz, which are near the boundaries of AZs of the traditional LPFs with I-type and II-type unit cells. Figure 7 shows the displacement amplitude of nodes along the \(z\) direction of the CLPF under excitation frequencies \(f_{1}\) = 4.07 Hz and \(f_{2}\) = 6.41 Hz. It should be noted that \(f_{1}\) = 4.07 Hz is in the first AZ of traditional LPF with II-type unit cells but outside the AZs of traditional LPF with I-type unit cells. It can be seen from Figure 7 that for \(f_{1}\) = 4.07 Hz, deformation of the rubber layers of the CLPF increases propagating in I-type unit cells and then decreases in II-type unit cells in the CLPF, which leads to a slight amplification of the displacement amplitude of the top layer compared with that of the bottom layer. On the other hand, \(f_{2}\) = 6.41 Hz is in both the first AZs of the traditional LPFs with I-type and II-type unit cells. Thus, in Figure 7, it can be seen that the displacement amplitude is significantly attenuated along the \(z\) direction for \(f_{2}\) = 6.41 Hz. Figure 8 shows the LBF and UBF of the first CAZ (i.e., union of the AZs of the traditional LPFs with I-type and II-type unit cells) and those of the frequency range of negative FRFs for 20 CLPFs with various filling ratios, where LBF\({}_{1}\), UBF\({}_{1}\), LBF\({}_{2}\) and UBF\({}_{2}\) represent the lower bound frequency and upper bound frequency of the first CAZ and those of the frequency range of negative FRFs, respectively. It can be seen from Figure 8 that the relative errors between LBF\({}_{1}\) and LBF\({}_{2}\) are less than 10% and those between UBF\({}_{1}\) and UBF\({}_{2}\) can be neglected, which verifies that the computation for the first CAZ simply by the union of the first AZs of the traditional LPFs with I-type and II-type unit cells is sufficiently accurate in the engineering field. Figure 5: AZs of traditional layered periodic foundations with (**a**) I-type and (**b**) II-type unit cells changing with the filling ratio of rubber material. Figure 8: (**a**) LBF and (**b**) UBF of the first CAZ and those of the frequency range of negative FRFs. Figure 6: FRF of the combined layered periodic foundation with \(N_{1}=2\) and \(N_{2}=2\). (The geometric parameters of I-type unit cell are \(a=0.5\,\mathrm{m}\) and \(f_{r}=30\%\), and those of \(\Pi\)–type unit cell are \(b=1\,\mathrm{m}\) and \(f_{r}=30\%\)). Figure 7: Displacement amplitude of nodes along the \(z\) direction of the combined layered periodic foundation under excitation frequencies \(f_{1}\) and \(f_{2}\). Figure 9 shows the FRFs of the CLPF with various numbers of I-type and II-type unit cells. It can be seen from Figure 9 that the excitation attenuation in the CAZs is larger as the numbers of I-type and II-type unit cells in the CLPF increases, which is in accordance with the results in the previous studies of traditional LPFs [18]. Figure 10 shows the FRFs of the CLPF with two I-type unit cells and various numbers of II-type unit cells.