## 4 Procedures
### Case Studies
#### 4.2.2 Numerical Modeling of a Dynamic Structure
 \tag{35}\]  
In Equation (35), the receptance matrix for each substructure i (A,B,C) is calculated from \([\text{H}^{\text{i}}]=\left[-\omega^{2}[\text{M}^{\text{i}}]+[\text{K}^{ \text{i}}]\right]^{-1}\), where \([\text{M}^{\text{i}}]\) and \([\text{K}^{\text{i}}]\) are the mass and stiffness matrices of substructure i, derived from the beam element formulation in Equations (36) and (37). Figure 6(c) shows the dynamic structure, consisting of three substructures modeled using beam elements as well as a substructuring scheme and the interested generalized coordinates and DoFs, while Table 1 summarizes the material properties and structural parameters used for simulation. Validation of the GRC-FBS Method  
To validate the GRC-FBS method, its results are compared to those obtained from the numerical FEA and modal analysis. The receptance matrix from the numerical FEA is calculated by modeling each substructure individually. To ensure consistent DoFs across all methods, each substructure is represented as a single beam element despite the higher discretization typically required for beam models. The subsystems A, B, and C are modeled using their stiffness and mass matrices \([\text{K}^{e}]\) and \([\text{M}^{e}]\) for all three beam elements, as expressed in Equations (35) and (36) [33; 34]. This simplification ensures consistency and reduces computational complexity. The local stiffness and mass matrices for each substructure are defined as follows:  
\[[\text{K}^{e}]=\frac{\text{EI}_{\text{e}}}{\text{L}_{\text{e}}^{3}} \begin{bmatrix} 12&6\text{L}_{\text{e}}&-12&6\text{L}_{\text{e}}\\ 6\text{L}_{\text{e}}&4\text{L}_{\text{e}}^{2}&-6\text{L}_{\text{e}}&2\text{L}_ {\text{e}}^{2}\\ -12&-6\text{L}_{\text{e}}&12&-6\text{L}_{\text{e}}\\ 6\text{L}_{\text{e}}&2\text{L}_{\text{e}}^{2}&-6\text{L}_{\text{e}}&4\text{L}_ {\text{e}}^{2}\end{bmatrix}  \tag{36}\]  
\[[\text{M}^{e}]=\frac{\text{m}_{\text{e}}}{420} \begin{bmatrix}156&22\text{L}_ {\text{e}}&54&-13\text{L}_{\text{e}}\\ 22\text{L}_{\text{e}}&4\text{L}_{\text{e}}^{2}&13\text{L}_{\text{e}}&-3\text{L}_ {\text{e}}^{2}\\ 54&13\text{L}_{\text{e}}&156&-22\text{L}_{\text{e}}\\ -13\text{L}_{\text{e}}&-3\text{L}_{\text{e}}^{2}&-22\text{L}_{\text{e}}&4\text{ L}_{\text{e}}^{2}\end{bmatrix}  \tag{37}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Modulus of** & **Moment of** & **Length of Beam** & **Cross-Sectional** & **Density (\(p\))** \\
**Elasticity (E)** & **Inertia (I)** & **Element (L)** & **Area (A)** & **Density (\(p\))** \\ \hline Data & 210 \(\times\) 109 Pa & 1 \(\times\) 10\({}^{-6}\) m\({}^{4}\) & 2 m & 0.01 m\({}^{2}\) & 7850 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties and structural parameters used for modeling. The global receptance matrix \([\mathrm{H}^{\mathrm{ABC}}]\) of the assembly system AB, is computed as follows:  
\[[\mathrm{H}^{\mathrm{ABC}}]=[-\omega^{2}[\mathrm{M}^{\mathrm{ABC}}]+[\mathrm{K}^{ \mathrm{ABC}}]]^{-1} \tag{38}\]  
where \([\mathrm{M}^{\mathrm{ABC}}]\) and \([\mathrm{K}^{\mathrm{ABC}}]\) are the global mass and stiffness matrices, which are calculated by combining the stiffness and mass matrices of the three elements. For the modal analysis, assuming an undamped system, the components of the receptance matrix \(\mathrm{H}_{\mathrm{ij}}(\omega)\) are determined using [35]:  
\[\mathrm{H}_{\mathrm{ij}}(\omega)=\frac{\mathrm{X}_{\mathrm{i}}}{\mathrm{F}_{ \mathrm{j}}}=\sum_{\mathrm{r}=1}^{\mathrm{n}}\frac{\phi_{\mathrm{ir}}\phi_{ \mathrm{ir}}}{(\omega_{\mathrm{r}}{}^{2}-\omega^{2})} \tag{39}\]  
Here, \(\omega_{\mathrm{r}}\) is natural frequency of the r-th mode, \(\phi_{\mathrm{ir}}\) is the normalized mode shape at \(\omega_{\mathrm{r}}\), and n represents total number of DoFs or modes. Given the configuration of the problem--three elements and four nodes, each with rotational and translational DoFs--the system has eight DoFs.