## 3 Results and Discussion  
For all model simulations, the material properties and dimensions adopted in this work are those used by both [PERSON] et al. [5] and [PERSON] et al. [18]. Starting with a single horizontal beam metamaterial structure with eight unit cells (resonators) and analyzing Equations (1)-(3) numerically and substituting the results into Equation (6), yields the FRF transmissibility shown in Figure 5.  
To generate the transmissibility in Figure 5, the frequency was swept from 0 to 500 Hz. This is because low frequency mechanical vibrations commonly found in nature are less than 300 Hz [21]. The input signal from the vibration source was taken from the left end of the horizontal beam (\(x=0\)), while the output signal was taken from the right end of the horizontal beam (\(x=L\)). The results show the presence of one prominent frequency bandgap, i.e., 300-415 Hz. This bandgap surrounds the resonant frequency of the local resonator, i.e., 309 Hz. The results shown in Figure 5 reveal that the local resonators trap these vibrations within this frequency bandgap, making the bandgap a local-resonator-influenced bandgap. The bandgap in Figure 5 agrees with the frequency bandgap reported and experimentally validated by [PERSON] et al. [5]. Moreover, using the same material dimensions and material properties reported in Table 1, a similar local resonator bandgap was reported by [PERSON] et al. [18].  
To study the behavior of the bandgap generated by the double-layered beam-type metamaterial structure in Figure 4A, the Runge-Kutta numerical method was used to analyze Equations (6), (16) and (20). The resulting FRF of the transmissibility, generated using Equation (21), is shown in Figure 6.  
The results from Figure 6 show two major bandgaps at (238-275 Hz) and (300-410 Hz). These two major bandgaps are marked by the shaded areas in Figure 6. The first important bandgap occurred at 300-410 Hz. This bandgap is maintained within the resonant frequency region of the local resonators, i.e., local-resonator bandgap. In addition to this fundamental frequency bandgap, an extra significant bandgap in the lower frequency region appears in Figure 6, i.e., 238-275 Hz. This second bandgap is the result of the [PERSON]'s scattering phenomena [32]. In effect, with double-layering, the metamaterial beam structure showed the ability to generate multiple bandgaps and at a lower frequency level. Having these multiple bandgaps with some of them happening at lower frequency is desirable for practical applications of metamaterial structures [21].  
Figure 5: Transmissibility bandgap of a single-horizontal beam metamaterial with 8 unit cells. The material and design parameters were taken from Table 1. Results show a significant frequency bandgap at 300–415 Hz.  
Comparing the results from Figures 5 and 6, the advantage offered by a double-layered beam-type metamaterial structure over a single-beam metamaterial structure is evident. First, Figure 6 shows a clear additional lower frequency bandgap in double-layered metamaterial structure, i.e., 238-275 Hz. Moreover, Figure 6 exhibits two frequency bandgaps, i.e., 238-275 Hz and 300-410 Hz, for the double-layered metamaterial structure as compared with a single frequency bandgap, i.e., 300-415 Hz in the single-beam metamaterial structure shown in Figure 5. This is particularly useful in a situation where a specific frequency range is required while an extra bandgap is also desirable. It is worth noting that, in [PERSON] et al. [18], a low frequency bandgap in the single-beam metamaterial was achievable using the negative stiffness design adopted in their work. This negative stiffness resulted in only a single frequency bandgap. Thus, a main advantage of the double-layered beam-type metamaterial structure presented in this article is the ability to create a low frequency bandgap in addition to the local-resonator frequency bandgap, as shown in Figure 6.