## 3 Case Studies: Application to Continuous and Lumped-Parameters Systems
### Case Study 2: Coupling of Subsystems with Lumped Parameters
#### 3.2.1 RCFBS Method and Governing Equations  
This example provides fundamental details for determining the receptance components of a system denoted as AB, which consists of four generalized coordinates, each with one translational degree of freedom (DoF), resulting in a total of four DoFs. The system is decomposed into two subsystems, A and B, each having three generalized coordinates with one translational DoF. Figure 6 illustrates the original assembly system AB and its decomposition into independent subsystems A and B. During the partitioning, mass m\({}_{25}\) is divided into masses m\({}_{2}\) and m\({}_{5}\), while mass m\({}_{36}\) is split into masses m\({}_{3}\) and m\({}_{6}\). Each partitioned mass is assigned to a different subsystem, with one part belonging to subsystem A and the other to subsystem B. The connection between the subsystems is established through two rigid connection points along the partition line. Both subsystems consist of multiple lumped masses, springs, and dampers. Unlike the previous case, the inclusion of internal DoFs within these subsystems adds complexity, which is effectively managed using the RCFBS method. Figure 6 shows the assembly system AB, Figure 6 presents the generalized coordinates at internal and connection points, and Figure 6 illustrates the substructuring scheme for decomposing system AB into subsystems A and B. The acceptance components are first calculated using the RCFBS method. These findings are subsequently validated by comparing them with the direct method and the modal method. The derivation of the receptance elements begins by assigning generalized coordinates and relevant DoFs based on boundary conditions, using the receptance coupling method based on FBS. The RCFBS method involves solving the equations of motion by maintaining force equilibrium and ensuring compatibility at the points of connection. Once these equations are solved, the receptance matrix HAB is obtained. Additionally, the receptance matrix components for the assembly system AB are presented in terms of the substructures' receptance using the generalized receptance coupling frequency-based substructuring (GRCFBS) formula, which is more effective than classical RCFBS formulations for complex systems. The validation of this case study is further supported by a comparison with the direct and modal methods. The equations of motions for substructure A and B, based on the generalized coordinates shown in Figure 6b, are expressed as follows:  
\[ \begin{bmatrix}\mathrm{H}_{\mathrm{u}_{1}\mathrm{u}_{1}}^{\mathrm{A}}&\mathrm{ H}_{\mathrm{u}_{1}\mathrm{u}_{1}\mathrm{c}_{1}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{u}_{1} \mathrm{u}_{1}\mathrm{u}_{2}}^{\mathrm{A}}\\ \mathrm{H}_{\mathrm{u}_{1}\mathrm{c}_{1}\mathrm{u}_{1}}^{\mathrm{A}}&\mathrm{ H}_{\mathrm{u}_{1}\mathrm{u}_{1}\mathrm{c}_{1}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{u}_{1} \mathrm{u}_{1}\mathrm{c}_{2}}^{\mathrm{A}}\\ \mathrm{H}_{\mathrm{u}_{2}\mathrm{c}_{2}\mathrm{u}_{1}}^{\mathrm{A}}&\mathrm{H} _{\mathrm{u}_{2}\mathrm{u}_{1}\mathrm{c}_{1}}^{\mathrm{A}}&\mathrm{H}_{\mathrm{ u}_{2}\mathrm{u}_{2}\mathrm{c}_{2}}^{\mathrm{A}}\end{bmatrix}\begin{cases} \mathrm{F}_{\mathrm{u}1}^{\mathrm{A}}\\ \mathrm{P}_{\mathrm{u}1}^{\mathrm{A}}\\ \mathrm{P}_{\mathrm{u}2}^{\mathrm{A}}\end{cases} = \begin{cases}\mathrm{u}_{ \mathrm{u}}^{\mathrm{A}}\\ \mathrm{u}_{\mathrm{u}\mathrm{u}\mathrm{c}1}^{\mathrm{A}}\\ \mathrm{u}_{\mathrm{u}\mathrm{c}2}^{\mathrm{A}}\end{cases} ,\\ \mathrm{H}_{\mathrm{u}_{\mathrm{u}\mathrm{c}2}}^{\mathrm{B}}\end{cases} \tag{21}\]  
The boundary conditions, including equilibrium and compatibility, are given by the following:  
\[\left\{ \begin{aligned} \mathrm{F}_{\mathrm{u}1}^{\mathrm{e} \mathrm{t}}^{\mathrm{A}}=\mathrm{F}_{\mathrm{u}1}^{\mathrm{A}}+\mathrm{F}_{ \mathrm{u}1}^{\mathrm{B}}\\ \mathrm{F}_{\mathrm{u}1}^{\mathrm{A}}=\mathrm{K}_{1}\left(\mathrm{u}_{\mathrm{ u}\mathrm{u}\mathrm{c}1}^{\mathrm{A}}-\mathrm{u}_{\mathrm{u}\mathrm{c}1}^{\mathrm{B}} \right)^{2}\end{aligned} \right. ;\left\{