## 4 Results and Discussion
### Case Study 1: Flexible Beam Coupling
#### 4.1.1 Direct Receptance Components at Node with Generalized Coordinate \(\mathcal{U}_{1}\)
In Figure 10, the receptance component \(H_{\theta_{1}\theta_{2}}^{AB}\), representing the rotational displacement at the tip due to a rotational moment applied at the same location, shows a strong alignment between the GRCFBS and FEA methods across all frequencies. Both methods accurately capture the resonance and anti-resonance peaks, suggesting that the EBT is effective in modeling the system's rotational dynamics within the frequency range considered. This alignment is particularly notable because rotational displacement under a rotational moment is well captured, indicating that EBT sufficiently models the bending resistance and rotational behavior of the system, which dominates in this frequency range. When comparing this result with Figures 8 and 9, the key difference emerges in how each method handles translational displacement responses. In Figure 8, which represents translational displacement due to a translational force, both methods align well up to approximately 2000 Hz, but discrepancies start to appear as the frequency increases, primarily due to limitations in accurately capturing high-frequency inertia effects and shear deformation. Similarly, in Figure 9, which shows the translational displacement response due to a rotational moment, both methods match closely in the primary modes, but a slight frequency shift in FEA suggests a softer structure with increased deformation amplitude, especially at higher frequencies. However, in Figure 10, the rotational displacement response under a rotational moment does not show such discrepancies, and both methods remain in strong agreement across the entire frequency range, even beyond 2000 Hz. This observation is significant because it highlights the different behavior of rotational dynamics compared to translational dynamics. Rotational dynamics, such as the response captured in Figure 10, are less influenced by higher-order effects like inertia and shear deformation, which primarily affect translational displacement responses (as seen in Figures 8 and 9). Thus, while Figures 8 and 9 show discrepancies between the GRCFBS and FEA methods at higher frequencies due to the limitations of both models in capturing inertia effects and shear deformation, Figure 10 demonstrates a consistent alignment in rotational displacement responses, suggesting that the EBT is well-suited for modeling rotational behavior even at higher frequencies. This reinforces the idea that while translational dynamics are more sensitive to high-frequency phenomena, the rotational dynamics remain well-modeled across the frequency range considered. A comparison of the receptance components \(H_{y_{1}y_{1}}^{AB}\), \(H_{y_{1}\theta_{1}}^{AB}\), and \(H_{\theta_{1}\theta_{1}}^{AB}\) over an expanded frequency range up to 6000 Hz, covering all four modes, is presented in Figure 11. Despite differences in values arising from the specific nature of the excitation and response, resonance frequencies remain consistent across various excitation types, reinforcing their status as intrinsic system characteristics. This consistency indicates that the resonance frequencies predicted by the different excitations--whether translational force or rotational moment-- remain consistent. Such alignment underscores that natural frequencies are inherent properties of the system and remain unaffected by the type of excitation applied.