## 3 Case Studies: Application to Continuous and Lumped-Parameters Systems
### Equations of motion for subsystem B:
#### 3.2.4 Results: Comparison of Receptors Components Across Different Methods  
Following the methodology outlined in the previous case study, the most straightforward approach for validating the results obtained from the receptance coupling and frequency-based substructuring (RCFBS) method is through a direct comparison with results from other established methods. In this study, the direct method, representative of an exact solution, and the modal method were employed for this purpose. A comparative simulation was conducted in MATLAB (2023b) using the theoretical approaches and parameters discussed earlier. The results, as shown in Figure 7, demonstrate a strong correlation between the RCFBS method and the other methods, particularly in terms of resonance peaks and natural frequencies. Specifically, the natural frequencies obtained via the RCFBS method for the four modes are 1.416, 3.25, 4.64, and 5.48 Hz. The corresponding frequencies from the direct and modal methods are 1.410, 3.177, 4.565, and 5.396 Hz, respectively. The differences are minimal, with deviations of 0.4%, 2%, 1.7%, and 1.5%, respectively. This slight increase in natural frequencies observed in the RCFBS method suggests a higher system stiffness, which is also reflected in the increased peak magnitudes. Beyond resonance peaks, there is a notable consistency between the methods in the anti-resonance regions as well, indicating that the dynamic response characteristics are reliably captured across the different approaches. In the context of this lumped-parameter system, the modal method does not suffer from mode truncation, unlike in continuous or distributed-parameter systems discussed in the previous case study. As a result, all methods are based on the same degrees of freedom (DoFs), leading to a more precise alignment in the results.  
As anticipated, all receptance components exhibit similar resonance peaks, which validates the accuracy of the results, given that natural frequencies are intrinsic properties of the system and are independent of the excitation point. Additionally, there is a strong agreement between both the cross-acceptance components (H\({}_{12}\), H\({}_{13}\), H\({}_{14}\), and H\({}_{23}\)) and the direct receptance components (H\({}_{11}\), H\({}_{22}\), H\({}_{33}\), and H\({}_{44}\)). Synthesizing the findings from the two different case studies, it is evident that the RCFBS method provides a reliable and effective solution for the reduced-order modeling of dynamic structures, showing comparable results to other methods. Importantly, RCFBS demonstrates superiority over the modal method in distributed-parameter systems due to its avoidance of mode truncation limitations.  
It is worth noting that the partitioning scheme employed in this study was based on trimming two masses associated with generalized coordinates u\({}_{2}\) and u\({}_{3}\) (masses m\({}_{25}\) and m\({}_{36}\)) to create a rigid connection. However, alternative partitioning schemes could be selected depending on specific objectives and points of interest. For instance, a different approach might involve using a flexible connection with springs and dampers instead of a rigid connection, which, while not explored in this paper, could yield similar results and provide further insights.  
Figure 7: Comparison between three methods for determining receptance components of the system AB.