## 4 Results and Discussion
### Case Study 1: Flexible Beam Coupling
#### 4.1.1 Direct Receptance Components at Node with Generalized Coordinate \(\mathcal{U}_{1}\)  
The direct components of receptance matrix \([H^{AB}]\) of the coupled system AB at the tip node with generalized coordinate \(\mathcal{U}_{1}\)--\(H^{AB}_{y_{1}y_{1}}\), \(H^{AB}_{y_{1}\theta_{1}}\), and \(H^{AB}_{\theta_{1}\theta_{1}}\)--are represented in Figures 8-10, comparing the magnitudes of these components. Notably, \(H^{AB}_{\theta_{1}y_{1}}\) is omitted due to reciprocity, i.e., \(H^{AB}_{\theta_{1}y_{1}}=H^{AB}_{y_{1}\theta_{1}}\). Figure 8 shows \(H^{AB}_{y_{1}y_{1}}\), the translational displacement at the tip caused by a translational excitation force applied at the same location. The FEA and GRCFBS results closely match for the primary two natural frequencies. For the first mode, FEA estimates the first mode at 368.1 Hz and GRCFBS at 383 Hz, a 3.8% increase, reflecting the stiffer response captured by the GRCFBS method. For the second mode, FEA predicts 1020.9 Hz, while GRCFBS gives 1025.7 Hz, resulting in a minor 0.4% difference, which is much less significant. Despite these small discrepancies in natural frequencies, it is important to highlight that the GRCFBS results show slightly reduced amplitude at dominant modes compared to FEA. This can be attributed to the continuous (distributed) nature of the system of the system in GRCFBS, which inherently suggests a stiffer system and, therefore, results in a reduced response amplitude. In the context of the EBT used in the GRCFBS method, each beam is treated as a continuous element, providing a simplified solution that may not fully account for flexibility  
Figure 11: Comparison between direct/cross-receptance components at tip (\(H^{AB}_{y_{1}y_{1}}\), \(H^{AB}_{y_{1}\theta_{1}}\), \(H^{AB}_{\theta_{1}\theta_{1}}\)) (case study 1). Figure 10: Acceptance component (\(H^{AB}_{\theta_{1}\theta_{1}}\)): rotational displacement at the tip due to couple at the same point (case study 1). at higher frequencies. Given that the system (AB) has four DoFs, four peaks are expected, but only two natural frequencies fall within the range up to 3000 Hz. Near 3000 Hz, GRCFBS shows an anti-resonance not captured by FEA. Up to 2000 Hz, the receptance component--i.e., the relationship between the applied force and resulting displacement--shows good agreement between FEA and GRCFBS, particularly in the lower-frequency regime where both methods perform similarly. However, as the frequency increases beyond 2000 Hz, discrepancies begin to grow, primarily due to the limitations of the beam model theory used in both numerical FEA and GRCFBS at high frequencies. At these higher frequencies, inertial effects and material damping become increasingly important, but are not fully accounted for in the current model, leading to increased divergence between the two methods. Figure 9 displays the receptance component \(H_{y_{1}\theta_{1}}^{AB}\), showing the translational displacement at the tip resulting from a couple applied at the same location. Both methods demonstrate similar trends in the resonance and anti-resonance peaks of the receptance component \(H_{y_{1}y_{1}}^{AB}\), as illustrated in Figure 8. Notably, a frequency shift is observed beyond 2000 Hz. This shift indicates an increase in stiffness in the FEA response, likely due to higher-order effects that are more pronounced in FEA. However, it is important to note that both methods encounter limitations when modeling high-frequency dynamics, where inertia effects become increasingly dominant. These effects, including shear deformation and rotational inertia, are not adequately captured by the assumptions in EBT, which is used in the GRCFBS method. Similarly, the beam models employed in FEA--which rely on stiffness and mass matrices and the coupling between stiffness and mass at nodes--also fail to fully account for these higher-frequency phenomena. As a result, both methods may misrepresent the system's true behavior at these elevated frequencies, particularly where inertia effects play a significant role.