## 4 Results and Discussion
### Case Study 1: Flexible Beam Coupling
#### 4.1.2 Direct Acceptance Components at Connection Point (Node \(\mathcal{U}_{2}\) )  
Understanding the receptance components at the connection point is crucial for assessing the dynamic behavior of the system, particularly where interactions between substructures occur. These components form a critical part of the overall receptance matrix, as they effectively capture the coupling dynamics between substructures. The accuracy of the connection receptance matrix, however, can be highly sensitive to perturbations such as noise or numerical errors. This sensitivity arises from the inverse operations inherent in the generalized coupling formula, as well as the receptance properties at the connection points, which are influenced by the individual substructures' dynamic characteristics. Accurately modeling these receptance components is, therefore, essential for ensuring reliable predictions of the system's overall behavior. In Figure 12, the receptance component \(H_{y_{2}y_{2}}^{AB}\) is presented, representing the translational displacement at the connection point caused by an applied excitation force at the same location. The results demonstrate a similar trend and level of accuracy observed in the tip node \(H_{y_{1}y_{1}}^{AB}\) (Figure 8) between the GRCFBS and FEA methods. This suggests that, even at the critical point of connection between two substructures, both methods can effectively capture the primary translational dynamics within the system. This consistency may arise from how well the compatibility and equilibrium boundary conditions at the connection point are incorporated into the GRCFBS approach.. Furthermore, the translational dynamics at the connection point are less influenced by higher-order effects like rotational coupling which can be observed in the result. Figure 13 illustrates the cross-acceptance component \(H_{y_{2}y_{2}}^{AB}\), which represents the translational displacement induced by applying a rotational moment at the connection  
Figure 12: Receptance (\(H_{y_{2}y_{2}}^{AB}\)): translational disp. at connection due to excitation force at same point (case study 1). point. Interestingly, this cross-receptance exhibits an even stronger correlation between the methods than was observed for the corresponding tip response in \(H^{AB}_{y_{1}\theta_{1}}\) (Figure 9). Both methods effectively predict resonance and anti-resonance peaks within the 0-3000 Hz range, demonstrating robust performance in modeling the coupling dynamics between translational and rotational motions. Figure 14 illustrates the receptance component \(H^{AB}_{\theta_{2}\theta_{2}}\) at the connection point, representing the rotational displacement caused by an applied excitation couple at the same location. The correlation between the GRCFBS and FEA methods is robust, closely aligning with the results observed for the tip node \(H^{AB}_{\theta_{1}\theta_{2}}\) (Figure 10). This strong agreement highlights the ability of both methods to effectively capture the rotational dynamics at the connection point, a critical interface where the coupling of substructures can influence system behavior. The similarity in trends suggests that the EBT (as used in GRCFBS) is capable of accurately modeling rotational dynamics at these connection points within the analyzed frequency range. Additionally, this consistency underscores the reliability of the GRCFBS approach in handling rotational degrees of freedom, even in regions where the interaction between substructures could introduce additional complexities. Figure 14: Receptance (\(H^{AB}_{\theta_{1}\theta_{2}}\)): Rotational disp. at connection point due to excitation couple at the same point (case study 1). Figure 13: Receptance (\(H^{AB}_{y_{2}\theta_{2}}\)): translational disp. at connection point due to excitation couple at same point (case study 1). Finally, Figure 15 compares \(H^{AB}_{y_{2}y_{2}},\ H^{AB}_{y_{2}\theta_{2}},\ \text{and}\ H^{AB}_{\theta_{2}\theta_{2}}\) across a higher frequency range up to 6000 Hz to capture all four modes.