## 5 Results  
This section presents a comparative analysis of the receptance matrix components for the assembly system, derived using three different methods. The focus is on evaluating the accuracy and reliability of the GRCFBS against numerical FEA and modal analysis. The results are depicted over a frequency range covering all modes, highlighting the behavior of direct and cross receptance components. The magnitudes of the direct and cross components of the assembly system's receptance matrix \(\left[\mathrm{H}^{\mathrm{ABC}}\right]\) are presented in Figures 8-15. Due to the inherent symmetry of the system, the receptance matrix components H\({}_{44}\) and H\({}_{11}\) exhibit similar behavior. Likewise, H\({}_{33}\) is identical to H\({}_{22}\) for the same reason. It compares the receptance components over a frequency range up to 160 Hz, covering all modes. Despite variations in values due to the excitation and response specifics, all three methods align well with the peaks, indicating that resonance frequencies are inherent properties of the system, independent of excitation type. The peaks observed in Figures 8-15 correspond to the natural frequencies of the system, representing resonances where the system exhibits significant dynamic responses. Each peak signifies a mode of vibration that arises from the assembly's structural properties, such as stiffness and mass distribution. For example, the lower-frequency peaks are generally associated with rigid body modes, where the system's components move as a whole. Conversely, higher-frequency peaks represent flexible modes, indicating localized deformations within the structure. As summarized in Table 2 for the system ABC with eight DoFs, the lower modes (modes 1 and 2) represent rigid body motion and are observed at 0 Hz, as expected for a free-free system without constraints. The subsequent peaks correspond to flexible modes at 5.13 Hz, 14.17 Hz, 31.09 Hz, 55.81 Hz, 108.06 Hz, and 132.94 Hz. These frequencies match  
Figure 9 compares the receptance component \(\rm H_{3}^{ABC}\), representing the translational displacement at node 1 due to an excitation force applied at the same point. The results demonstrate a high degree of agreement across a frequency range covering six flexible modes. Figures 10 and 11 compare the receptance components \(\rm H_{y_{1}\theta_{1}}^{ABC}\) and \(\rm H_{y_{1}y_{1}}^{ABC}\), respectively. These measurements were conducted specifically to verify the reciprocity property inherent in the system. According to this property, these components are expected to be mathematically identical, as demonstrated in the figures. \(\rm H_{y_{1}\theta_{1}}^{ABC}\) represents the translational  
Figure 11: Comparison of the receptance component \(\rm H_{21}\) (\(\rm H_{\theta_{1}y_{1}}^{ABC}=H_{y_{1}\theta_{1}}^{ABC}\)) using three different methods. Figure 12: Comparison of the receptance component \(\rm H_{22}\) (\(\rm H_{\theta_{1}\theta_{1}}^{ABC}\)) using three different methods. Figure 13: Comparison of the receptance component \(\rm H_{13}\) (\(\rm H_{y_{1}y_{2}}^{ABC}\)) using three different methods. displacement at node 1 resulting from an excitation couple applied at the same point, while H\({}_{0.Y_{1}}^{\text{ABC}}\) represents the angular displacement at node 1 resulting from an excitation force applied at the same point. The identical appearance of the graphs serves as confirmation of the reciprocity property, validating the theoretical approach. This verification is critical for ensuring the robustness of the methodology across the frequency range. Figure 12 compares the receptance component H\({}_{0.\theta_{1}\theta_{2}}^{\text{ABC}}\), representing the rotational displacement at node 1 due to an excitation couple applied at the same point, and shows a strong alignment with the numerical FEA and modal analysis, verifying the effectiveness of the generalized coupling method. The resonant peaks corresponding to the flexible modes are accurately depicted by the GRC-FBS method, with natural frequencies deriving from the other two methods.