## 4 Finite Element Study
### Impact of Spacial Variations on Bandgap Characteristics  
The established methodology enables manipulation of the transfer function, thereby permitting exploration into how adjustments in the mass placement on a resonator affect bandgap traits, a key factor for refining bandgap properties within a closed-loop control system.  
This section explores adjusting resonator stiffness while maintaining constant mass, a method beneficial for heavy machinery applications where traditional piezoelectric solutions may fall short. Stiffness tuning, as opposed to piezoelectric adjustments, offers a more durable and practical solution for these demanding environments. The current study examines a conventional metastructure that does not incorporate internally coupled resonators. The resonators are of the cantilever type, with a mass that can be positioned along the length from the tip to the base. The specific parameters defining the metastructure and resonators are as follows: eight resonators (\(N_{r}=8\)), with the beam dimensions being 300 mm in length, 25 mm in width, and 3 mm in height. The material density is 2700 kg/m\({}^{3}\), and the modulus of elasticity is 69.5 GPa. The damping ratio of the structure and resonators is the same, at 0.01. An attached mass (\(m_{a}\)) of 3.8 g is placed at distances that vary from 20 to 57.3 mm along the resonator. The natural frequency of the resonator (\(\omega_{r}\)), when the attached mass is at the tip, is 32 Hz. This setup allows for an exploration of the resonator stiffness's impact on the bandgap properties of the metastructure.  
Figure 11 provides a 3D visualization of how the position of the attached mass along the length of a resonator affects the bandgap frequencies in a metastructure. The natural frequency at which the bandgap starts is denoted as \(\omega_{r}\), corresponding to the case when the mass is located at the tip of the resonator. The graph demonstrates that as the mass moves closer to the base of the resonator--decreasing \(\delta\)--the resonator's stiffness increases, leading to a rise in \(\omega_{r}\) and a subsequent shift of the bandgap towards higher frequencies.  
The contour plot in the x-y plane clearly depicts the bandgap's initiation at the initial natural frequency \(\omega_{r}\) when the mass is at the resonator's tip. From there, the bandgap expands and moves as the location of the mass changes. This shift is particularly crucial for applications requiring tunable vibration isolation, as it shows the potential to adjust the bandgap frequency by simply repositioning the resonator mass without altering the resonator or structure itself.  
The binary representation in Figure 12 illustrates the influence of the mass location along the resonator on the bandgap frequencies. With the bandgap depth limit set at a decibel ratio of output to input displacement of 0.2, the plot shows that when the attached mass is positioned at the tip of the resonator, the bandgap originates at the resonator frequency  
Figure 11: A 3D plot showing the shift in bandgap frequency related to mass positioning on the resonator, with delta (\(\delta\)) representing the mass location from the resonator’s tip to base.  
\(\omega_{r}\). The white areas in the binary representation correlate to the regions of significant transmittance reduction, effectively mapping the bandgap's presence and evolution as the mass moves closer to the resonator's base.