## 3 Results and Discussion  
The rectangular beam under investigation has the following dimensions: a length of 0.91 m (\(L_{m}\)), a width of 40 mm (\(w_{m}\)), and a height of 3 mm (\(l_{m}\)). The material used for the beam has a density of 2710 km per cubic meter (\(\rho_{m}\)) and a modulus of elasticity of 52 GPa (\(E_{m}\)). The beam is characterized by a damping ratio of 0.03 (\(\zeta_{m}\)), and the analysis considers a total of eight vibration modes (\(N_{m}\)). Each resonator (\(N_{r}\)) within the system has a mass of 80 g (\(m_{r}\)) and a spring constant of 380 kiloewtons per meter (\(k_{r}\)). The damping ratio for the resonators is also set at 0.03 (\(\zeta_{r}\)). Figure 5 depicts the dispersion curve of the internally coupled metamaterial beam (\(\kappa=k_{r}\)) using the plane wave expansion method. The target frequency corresponds to the resonator frequency. The diagram illustrates two bandgaps: the first is associated with the in-plane behavior of both resonators within each unit cell, while the second bandgap emerges due to the out-of-plane behavior of the two resonators in each unit cell. Here, \(G_{n}\) represents the wave vector number and \(a\) denotes the lattice size. Figure 6 illustrates the transmittance characteristics of a metamaterial beam with internal resonator coupling in terms of tip displacement relative to the base displacement in absolute coordinates. The presence of a common initial bandgap aligns with the theoretical expectations discussed earlier, assuming that all resonators resonate at the same frequency (\(\omega_{2r-1}=\omega_{2r}=\omega_{r}\)) and maintain identical phase relationships. In the case of the internally coupled metastructure, an additional bandgap is observed, which substantiates the theoretical premise that variations in resonator frequencies or phase differences can extend the parameter \(\kappa\). This extension, facilitated by the assumption of a massless coupling spring, introduces new dynamics to the system by adding an extra pole and zero, resulting in the creation of an additional bandgap. The primary bandgap occurs at the target frequency, which corresponds to the resonator's frequency adjusted by the length factor \(\sqrt{1+\mu}\). The secondary bandgap's location is contingent upon the stiffness of the internal coupling and is defined by the length factor \(\sqrt{1+\frac{\mu}{4}}\). Notably, the dips in the graph signify areas of low transmittance, indicating reduced vibration at the beam's tip and effectively marking the bandgap regions. Figure 5: Dispersion curve of an internally coupled metamaterial beam, displaying two distinct bandgaps resulting from in-plane and out-of-plane resonator behavior. Figure 7 presents a graphical analysis illustrating the influence of varying internal coupling spring constant values, denoted as \(\kappa\), on the bandgap frequencies within a metastructure. Notably, alterations in \(\kappa\) do not induce substantial shifts in the frequency edges of the primary first bandgap. However, as \(\kappa\) increases, it introduces additional, narrower gaps at frequencies above the rest of the second bandgap. These narrower gaps underscore the sensitivity of the metastructure's dynamic response to specific ranges of internal coupling strength. Figure 8 presents a contour plot of the transmittance across the metastructure as a function of the normalized internal coupling strength, \(\kappa/\omega_{r}\), and normalized frequency, \(\omega/\omega_{r}\). The color gradient represents the logarithmic scale of transmittance, indicating the level of wave attenuation within the metastructure. Dark regions correspond to high attenuation levels, signifying the presence of bandgaps. As observed, the contour lines delineate the boundaries of the bandgaps, which become more distinct with specific values of internal coupling strength.