## 4 Finite Element Study  
The dynamic behavior of metastructures incorporating internally coupled resonators is investigated using finite element method (FEM) simulations, affirming theoretical predictions. The analysis outputs present the vibrational modes of the metastructure. These modes are expressed as amplitude variations across a spectrum of normalized frequencies, with a particular focus on resonant frequencies pertinent to bandgap development. Significant findings from the analysis illustrate the variance in bandgap distribution and intensity of resonant peaks as a function of stiffness ratio. This implies a substantial relationship between internal coupling stiffness and the dynamic response of the metastructure. Visualization of these results not only confirms primary and secondary bandgap presence but also aligns with the theoretical implications of internal resonator coupling. Figure 9 concentrates on the transmissibility for a specific stiffness ratio \(\kappa\), reflecting a critical scenario where \(\kappa\) is precisely matched with the resonator's stiffness (\(\kappa/\omega_{r}=0.003\)), an essential condition for optimal bandgap definition. This particular observation underscores the necessity of accurate internal coupling stiffness to achieve the designed dynamic response.  
However, deviations from the ideal \(\kappa\) value lead to pronounced disorder within the system's response, emphasizing the metastructure's sensitivity to variations in internal coupling stiffness. Such irregularities pose challenges for ensuring predictability and  
Figure 8: Transmittance contour plot against normalized internal coupling strength and frequency, highlighting bandgap boundaries and the metastructure’s sensitivity to \(\kappa\) variations.  
consistent performance in practical applications, thus advocating for stringent precision in design and manufacturing processes.  
The contour plot depicted in Figure 10 utilizes a binary representation to mark regions of transmittance reduction, set at \(log(10^{-0.1})\). The provided binary representation displays two distinct white regions against a cyan background, illustrating the transmittance levels across various stiffness ratios \(\kappa\) and normalized frequencies. The first white region, located at the target resonator frequency \(\omega_{r}\), corresponds to a bandgap typically observed in conventional metamaterials. This bandgap represents a frequency range where the structure prevents wave propagation, thereby indicating a strong vibration isolation capability at the resonant frequency of the metamaterial. The second white region appears at a higher frequency range and signifies the impact of internal coupling within the metamaterial structure. This additional bandgap is a result of the specific design and internal resonator interactions that are a characteristic of the studied metastructures. The emergence of this second bandgap highlights the effect of internal coupling on the extension of vibration isolation performance to higher frequency ranges.  
Figure 10: Binary contour plot illustrating the presence and absence of transmittance corresponding to bandgaps as a function of stiffness ratio \(\kappa\) and normalized frequency.  
Figure 9: Transmissibility for a cantilever beam with stiffness ratio \(\kappa\) equal to \(k_{r}\), demonstrating optimal internal coupling for bandgap clarity.