## 4 Results and Discussion
### Case Study 1: Flexible Beam Coupling
#### 4.1.3 Cross-Receptance Components Between Tip and Connection Point  
Cross-receptance components reflect the dynamic coupling effects between different nodes in the system and play a pivotal role in validating the accuracy of the model's representation of system interactions. These components are essential for achieving a comprehensive understanding of the system's dynamic behavior. They assist in validating models, optimizing designs, and diagnosing performance characteristics by highlighting how forces or moments at one node influence displacements at another. The following figures illustrate the variations and correlations observed in these components. Figure 16 illustrates the cross-receptance component \(H^{AB}_{y_{2}y_{1}}\), which represents the translational displacement at the tip node caused by a force applied at the connection point. Unlike previous acceptance components, no anti-resonance is observed between the first two dominant resonant peaks. However, one anti-resonance appears after the second resonant peak. This behavior is effectively captured by the GRCFBS, the same as the FEA method, demonstrating strong alignment in characterizing this dynamic feature, particularly around the 2000 Hz frequency range. The absence of an anti-resonance between the first two dominant peaks can be attributed to the nature of the force transmission between the tip and the connection point in this specific cross-receptance case. Since the force is applied directly at the connection node, the coupling between the substructures is predominantly influenced by translational dynamics, which tend to smooth out higher-order oscillatory effects like anti-resonances in this frequency range. This results in a more direct interaction between the two resonant modes, without significant modal interference that would typically produce an anti-resonance. The appearance of an anti-resonance after the second resonant peak can be explained by the interaction of higher-order modes, where rotational and translational coupling effects become more pronounced. At this frequency, the dynamic behavior of the system introduces phase cancellations between the input force and the system's response, leading to the anti-resonance. Both the GRCFBS and FEA methods successfully capture this characteristic, reinforcing their ability to model complex dynamic interactions at the connection node. Additionally, the accuracy observed around the 2000 Hz range highlights the robustness of the GRCFBS method in matching the benchmark FEA approach for the primary modes of interest. The consistent alignment of natural frequencies and the ability to replicate anti-resonance features validate the GRCFBS methodology as a reliable tool for analyzing cross-receptance dynamics, even in scenarios with nuanced interactions like this one. Figures 17 and 18 illustrate the indirect receptance components \(H^{AB}_{y_{2}\theta_{1}}\) and \(H^{AB}_{\theta_{2}\theta_{1}}\), representing the translational and rotational displacements, respectively, at the connection node caused by an excitation couple at the tip. Both graphs exhibit trends similar to those observed in Figure 16, particularly in terms of the resonance and anti-resonance regimes. For \(H^{AB}_{y_{2}\theta_{1}}\), the interaction highlights how rotational excitation at the tip is transmitted to induce translational dynamics at the connection node. Likewise, \(H^{AB}_{\theta_{2}\theta_{1}}\) demonstrates the rotational response at the connection node due to the same tip excitation. The results indicate a very strong correlation between the GRCFBS and FEA methods across the entire frequency range analyzed, particularly up to 2000 Hz. This alignment suggests that both methods are highly effective in capturing the system's coupled dynamic response within the dominant frequency modes.