## 3 Case Studies
### Case Study 2: Coupling of a Continuous Substructure and a Rigid Component  
To further demonstrate the application of the GRCFBS approach, the second case study investigates the receptance matrix for an assembly system labeled as \"AC\". This system is composed of two distinct subsystems: a flexible substructure A and a rigid substructure C, as shown in Figure 5. Substructure A is the same cantilever beam from the first case study, featuring a free-fixed boundary condition, while substructure C is modeled as a rigid body. Both the general receptance method and numerical FEM are applied to solve this problem.  
In Figure 6, the decomposition of system AC into substructures A and C using the substructuring technique is shown. Notably, there is only one generalized coordinate at connection point 2, with two DoFs: rotational and translational.  
In comparing the two cases, case study 1 involves a combination of two subsystems (A and B) with two generalized coordinates: one at the interface and the other as an internal coordinate of subsystem B. In contrast, case study 2 involves only one generalized coordinate at the connection point from a substructuring perspective, with no internal coordinates. Due to the rigid nature of substructure C, it acts as a lumped mass connected at the interface coordinate \(\mathcal{U}_{2}\), exhibiting rigid body motion.  
The governing equation for substructure A remains the same as in Equations (22) and (23) from case study 1. The governing equation for substructure C is expressed as follows:  
\[\{\mathcal{U}_{2}^{c}\}=\begin{bmatrix}\mathcal{H}_{22}^{c}\end{bmatrix}\{F_ {2}^{c}\}\ \Rightarrow\ \begin{cases}x_{2}^{c}\end{cases}=\begin{bmatrix}H_{y_{2}y_{2}}^{c}&H_{y_{2} \theta_{2}}^{c}\\ H_{y_{2}y_{2}}^{c}&H_{\theta_{2}\theta_{2}}^{c}\end{cases}\begin{cases}F_{y_{2} }^{c}\\ F_{\theta_{2}}^{c}=M_{\theta_{2}}^{c}\end{cases}. \tag{35}\]  
where \([\mathcal{H}_{22}^{c}]\) represents the receptance matrix for substructure C, and \(F_{y_{2}}^{c}\) and \(F_{\theta_{2}}^{c}(M_{\theta_{2}}^{c})\) are the force and moment excitations applied. As illustrated in Figure 7, excitation loads (force and moment) are applied separately to substructure C to determine the translational and  
Figure 5: Schematic of system AC, consisting of substructures A and C.  
Figure 6: Decomposition of system AC into substructures A and C using substructuring strategy.  
angular displacements. The receptance components relating displacements to excitation loads are given by the following:  
\[H_{y_{2}y_{2}}^{c}=\frac{-1}{m_{B}\omega^{2}},\ H_{y_{2}\theta_{2}}^{c}=H_{\theta _{2}y_{2}}^{c}=0,\ H_{\theta_{2}\theta_{2}}^{c}=\frac{-1}{I_{c}\omega^{2}}. \tag{36}\]  
where \(I_{c}\) and \(m_{c}\) are the moment of inertia and mass of the rigid wheel as substructure C, respectively. When a harmonic force \(F_{y}^{c}e^{i\omega t}\) and \(M_{\theta}^{c}e^{i\omega t}\) are applied to this rigid component, the assembly system AC is excited. The motion of the free end of the beam can be analyzed by considering both the beam's flexural response and the inertia of the rigid body. Since substructure C has no internal DoFs or generalized coordinates, the coupling equation for calculating the receptance function of the assembly AC is determined from the receptance components at the connection point using generalized receptance coupling:  
\[\mathcal{H}_{22}^{AC}=\frac{\mathcal{U}_{2}^{AC}}{F_{2}^{AC}}=\frac{\mathcal{U} _{2}^{C}}{F_{2}^{C}}=\mathcal{H}_{22}^{A}\Big{(}\mathcal{H}_{22}^{A}+\mathcal{ H}_{22}^{C}\Big{)}^{-1}\mathcal{H}_{22}^{B}=\mathcal{H}_{22}^{A}-\mathcal{H}_{22}^{A} \Big{(}\mathcal{H}_{22}^{A}+\mathcal{H}_{22}^{C}\Big{)}^{-1}\mathcal{H}_{22}^ {A}. \tag{37}\]  
Here, the receptance matrix for system AC is a \(2\times 2\) matrix, expressed as follows:  
\[[H^{AC}]=\begin{bmatrix}H_{y_{1}y_{2}}^{AC}&H_{y_{1}\theta_{2}}^{AC}\\ H_{\theta_{2}y_{2}}^{AC}&H_{\theta_{2}\theta_{2}}^{AC}\end{bmatrix}. \tag{38}\]