## 4 Results and Discussion
### Case Study 2: Flexible Beam-Rigid Wheel Coupling  
In this case study, the receptance components of the assembly system AC, which couples a flexible beam with a rigid wheel, have been analyzed. The assembly system is illustrated in Figures 5 and 6. The primary objective is to evaluate how the interaction between the flexible and rigid substructures influences dynamic behavior and the receptance characteristics of the coupled system. In this configuration, the tip and the connection point (node \(\mathcal{U}_{2}\)) are coincident, eliminating the distinct \(H_{y,y_{1}}^{AC}\) component. The receptance components are calculated using the theoretical approaches outlined in Section 3.2. For this analysis, the stiffness of the rigid wheel is neglected, resulting in a global stiffness matrix for the assembly (AC) that remains consistent with that of substructure A. Substructure C incorporates an inertia matrix without any stiffness in FEA. However, in the GRCFBS method, substructure C incorporates an FRF matrix accounting for both translational and rotational inertia. The coupled system AC exhibits two dominant modes due to the inclusion of a single generalized coordinate with two DoFs. Figures 20-23 illustrate the receptance components for this assembly. Figure 20 presents the direct receptance component \(H_{\theta 2\theta 2}^{AC}\), representing the translational displacement at node U\({}_{2}\) due to an applied excitation force. The results indicate that the rigid wheel's lack of stiffness significantly reduces system stiffness, leading to a shift in natural frequencies and altering the dynamic response. The GRCFBS method predicts the first mode at 483 Hz, while the FEA method predicts it at 468 Hz, reflecting a slight increase in effective stiffness (approximately 3%). This alignment demonstrates strong agreement between the two methods for the first dominant mode. However, deviations are observed at higher frequencies, particularly beyond 2000 Hz, where the influence of the rigid component becomes increasingly pronounced. These higher-frequency discrepancies are attributed to the growing dominance of inertia effects and the role of stiffness at connection points in the GRCFBS method. The accuracy of the GRCFBS method hinges on the precision of receptance coupling at the interface, which is influenced by stiffness, mass, and damping parameters. When the stiffness contribution of one substructure is omitted, as in the rigid wheel model, inherent errors arise in predicting the system's dynamic response. These errors become more evident at higher frequencies, where the receptance coupling mechanism is less effective. In contrast, the inclusion of stiffness contributions at connection points, as in case study 1, enhances the correlation between GRCFBS and FEA, particularly for systems with significant stiffness  
Figure 23: Comparison between cross-receptance components (\(H_{\theta 2\theta 2}^{AC}\), \(H_{\eta 2\theta 2}^{AC}\), and \(H_{\theta 2\theta 2}^{AC}\)) for 2 nd case study. Figure 22: Acceptance component (\(H_{\theta 2\theta 2}^{AC}\)): rotational disp. to excitation couple (2 nd case study). at the connection interface. This underscores the critical role of stiffness in improving the accuracy of receptance coupling predictions. Figures 21 and 22 depict the cross-receptance components \(H_{y_{2}\theta_{2}}^{AC}\) and \(H_{\theta_{2}\theta_{2}}^{AC}\), capturing the coupling between translational and rotational dynamics at the coincident tip/connection point due to applying an excitation moment. These components highlight the role of rotational stiffness introduced by the rigid wheel. The presence of anti-resonances, unique to this flexible-rigid coupling, indicates constrained vibrational modes that affect energy transfer within the system. Like \(H_{y_{2}y_{2}}^{AC}\), there is good agreement between the two methods up to 2000 Hz for the first dominant mode, and the discrepancy appeared for the second mode at higher frequencies.