## 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
\begin{aligned} \mathrm{F}_{\mathrm{u}2}^{\mathrm{e}\mathrm{t}}^{\mathrm{e}\mathrm{t}}=\mathrm{F }_{\mathrm{u}2}^{\mathrm{A}}+\mathrm{F}_{\mathrm{u}2}^{\mathrm{B}}\\ \mathrm{F}_{\mathrm{u}2}^{\mathrm{A}}=\mathrm{K}_{2}\left(\mathrm{u}_{\mathrm{ u}\mathrm{c}2}^{\mathrm{A}}-\mathrm{u}_{\mathrm{u}\mathrm{c}2}^{\mathrm{B}}\right)^{2} \end{aligned} \right., \tag{22}\]  
where \(K_{1}\) and \(K_{2}\) represent the dynamic stiffness at the connection points. To determine the direct receptance \(H_{\mathrm{u}_{1}\mathrm{u}_{1}}^{AB}\), which is the displacement of system AB at coordinate \(u_{1}^{AB}\) due to an excitation applied at the same point (\(H_{\mathrm{u}_{1}\mathrm{u}_{1}}^{AB}=\frac{u_{1}^{AB}}{E_{\mathrm{u}1}^{ \mathrm{A}}}\)), we start by  
Figure 6: The Substructuring scheme of assembly system AB to two subsystems A and B. (**a**) Assembly system AB; (**b**) Generalized coordinates at internal and connection points; (**c**) Substructuring scheme illustrating the decomposing system AB into subsystems A and B.  
the reciprocity principle and the application of the chain rule, the full system receptance matrix, which consists of 16 elements, can be determined by determining the elements of one column or row from the substructures when they are disassembled. It is important to note that, when any receptance element like \(H_{u_{1}u_{1}}^{AB}\) approaches infinity, it indicates resonance in the assembled system and provides the natural frequency of the entire system. Consequently, the natural frequencies \(\omega_{1}{}^{2},\omega_{2}{}^{2},\omega_{3}{}^{2}\), and \(\omega_{4}{}^{2}\) can be directly determined from the frequency equation by setting \(S_{0}=0\). Additionally, the fundamental development using the RCFBS method can be expressed in the generalized form of Equation (1), which presents the coupling of two substructures under the discussed assumptions. \[ \begin{bmatrix}\mathrm{H}_{u_{1}u_{1}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{1}}^{ \mathrm{AB}}&\mathrm{H}_{u_{1}u_{3}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{4}}^{ \mathrm{AB}}\\ \mathrm{H}_{u_{1}u_{1}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{2}}^{\mathrm{AB}}& \mathrm{H}_{u_{1}u_{3}}^{\mathrm{AB}}&\mathrm{H}_{u_{2}u_{4}}^{\mathrm{AB}}\\ \mathrm{H}_{u_{1}u_{1}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{2}}^{\mathrm{AB}}& \mathrm{H}_{u_{1}u_{3}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{4}}^{\mathrm{AB}}\\ \mathrm{H}_{u_{1}u_{1}}^{\mathrm{AB}}&\mathrm{H}_{u_{1}u_{2}}^{\mathrm{AB}}& \mathrm{H}_{u_{1}u_{3}}^{\mathrm{AB}}&\mathrm{H}_{u_{2}u_{4}}^{\mathrm{AB}}\\ \end{bmatrix} \] \[= \begin{bmatrix}\mathrm{H}_{u_{1}u_{1}}^{\mathrm{A}}&\mathrm{H}_{u_{1} u_{1}}^{\mathrm{A}}&\mathrm{H}_{u_{1}u_{1}}^{\mathrm{A}}&\mathrm{H}_{u_{1}u_{2}}^{ \mathrm{A}}&0\\ \mathrm{H}_{u_{1}u_{1}}^{\mathrm{A}}&\mathrm{H}_{u_{1}u_{1}}^{\mathrm{A}}& \mathrm{H}_{u_{1}u_{2}}^{\mathrm{A}}&0\\ \mathrm{H}_{u_{2}u_{1}}^{\mathrm{A}}&\mathrm{H}_{u_{2}u_{2}}^{\mathrm{A}}& \mathrm{H}_{u_{2}u_{2}}^{\mathrm{A}}&0\\ 0&0&0&\mathrm{H}_{u_{1}u_{2}}^{\mathrm{B}}\\ \end{bmatrix}  \tag{34}\]  
To determine the receptance components of subsystems A and B in their disassembled conditions, it is essential to evaluate them as independent substructures separately. As illustrated in Figure 6b, when system AB is partitioned into subsystems A and B, there are two generalized coordinates, \(\mathrm{uc}_{1}\) and \(\mathrm{uc}_{2}\), at the interface or connection area. Equation (34) demonstrates that the receptance of system AB depends on the receptance matrices of subsystems A and B, which are determined through either measurement or numerical analysis. Depending on the complexity and availability of the data, the frequency response function (FRF) matrix of the substructures can be obtained via direct measurement, or analytical or numerical methods. In this case, an analytical method can be employed to derive the receptance (FRF) matrix components at the points of interest.